\section{Introduction}
\let\thefootnote\relax\footnote{\textit{Key words}: exceptional Lie superalgebra $\cdot$ covariant $\cdot$ spinor representation $\cdot$ octonions $\cdot$ Mathews identities.}
\let\thefootnote\relax\footnote{\textit{2020 Mathematics Subject Classification}: 17A75, 17B25, 17B60, 53C27.}

The space of binary cubics, a symplectic representation of the Lie algebra $\mathfrak{sl}(2,k)$, has particular symplectic properties \cite{Eisenstein1844}, \cite{SlupinskiStanton2012}. It admits three covariants, among the Hessien and the discriminant, satisfying remarkable geometric identities \cite{Mathews11}. This representation is an example of a larger class of representations sharing these properties: the special $\epsilon$-orthogonal representations of colour Lie algebras \cite{StantonSlupinski15}, \cite{Meyer19Kostant}. The terminology special comes from their role in symplectic geometry \cite{CS09}. A special $\epsilon$-orthogonal representation $V$ of a colour Lie algebra $\mathfrak{g}$ can be extended to define a colour Lie algebra of the form
\begin{equation*}
    \tilde{\gg}=\mathfrak{g}\oplus \mathfrak{sl}(2,k) \otimes V\otimes k^2.
\end{equation*}
In this way, special symplectic representations of Lie algebras give rise to Lie algebras and special orthogonal representations of Lie algebras give rise to Lie superalgebras.
\vspace{0.1cm}

In this paper, from an octonion algebra $\mathbb{O}$ over $k$, we show that
\begin{itemize}
    \item a one parameter family of $4$-dimensional representations of $\mathfrak{sl}(2,k)\times \mathfrak{sl}(2,k)$ ;
    \item the $7$-dimensional fundamental representation ${\rm Im}(\mathbb{O})$ of the Lie algebra ${\rm Der}(\mathbb{O})$ ;
    \item the $8$-dimensional spinor representation $\mathbb{O}$ of the Lie algebra $\mathfrak{so}({\rm Im}(\mathbb{O}))$
\end{itemize}
are special orthogonal representations and give rise to exceptional Lie superalgebras of type $D(2,1;\alpha)$, $G_3$ and $F_4$ (in the Kac notation \cite{VKac1975}). This is similar to various constructions from Sudbery \cite{Sudbery83}, Kamiya and Okubo \cite{KamiyaOkubo03} and Elduque \cite{Elduque04}.
\vspace{0.1cm}

We explicitly compute the covariants of these representations. In particular, we give formulae of the moment maps of ${\rm Im}(\mathbb{O})$ and $\mathbb{O}$ and we show that the trilinear covariant of ${\rm Im}(\mathbb{O})$ is, up to a constant, the associator. The quadrilinear covariant of ${\rm Im}(\mathbb{O})$ admits a decomposition into a sum of $7$ decomposable forms which naturally correspond to the $7$ lines of the Fano plane and the two maps of the first Mathews identity are, up to constants, the Hodge duals of the cross-product on ${\rm Im}(\mathbb{O})$. Then we give a decomposition of the quadrilinear covariant of $\mathbb{O}$ into a sum of $14$ decomposable forms which naturally correspond to the $14$ affine planes of the affine space $(\mathbb{Z}_2)^3$. The two maps of the first Mathews identity are, up to a constant, the Hodge duals of the trilinear covariant of $\mathbb{O}$ and the two maps of the second Mathews identity are, up to constants, the Hodge duals of the moment map of $\mathbb{O}$.

For special orthogonal representations associated to basic classical Lie superalgebras and interpretation of their covariants, see the appendix of \cite{Meyer19Kostant}.

\begin{comment}

$$\mathfrak{sl}(2,k)\oplus \mathfrak{sl}(2,k)\oplus \mathfrak{sl}(2,k) \oplus k^2\otimes k^2\otimes k^2.$$

$${\rm Der}(\mathbb{O})\oplus \mathfrak{sl}(2,k) \oplus {\rm Im}(\mathbb{O})\otimes k^2.$$
The trilinear covariant of ${\rm Im}(\mathbb{O})$ is, up to a constant, the associator and the quadrilinear covariant of ${\rm Im}(\mathbb{O})$ admits a decomposition into a sum of $7$ decomposable forms which naturally correspond to the $7$ lines of a Fano plane. We show that the two maps of the first Mathews identity are, up to constants, the Hodge duals of the cross-product on ${\rm Im}(\mathbb{O})$.

We show that the $8$-dimensional spinor representation $\mathbb{O}$ of the Lie algebra $\mathfrak{so}({\rm Im}(\mathbb{O}))$ is special orthogonal in Section \ref{section F4}. The corresponding Lie superalgebra is an exceptional simple Lie superalgebra of type $F_4$ and of the form:
$$\mathfrak{so}({\rm Im}(\mathbb{O}))\oplus \mathfrak{sl}(2,k) \oplus \mathbb{O}\otimes k^2.$$
Then we give a decomposition of the quadrilinear covariant of this representation into a sum of $14$ decomposable forms which naturally correspond to the $14$ affine planes of an affine space of dimension $3$ on $\mathbb{Z}_2$. The two maps of the first Mathews identity are, up to a constant, the Hodge duals of the trilinear covariant of $\mathbb{O}$ and the two maps of the second Mathews identity are, up to constants, the Hodge duals of the moment map of $\mathbb{O}$.
\end{comment}

\subsection*{Acknowledgements} The author wants to express his gratitude to Marcus J. Slupinski for his suggestions and his encouragement. This work was partially supported by the Engineering and Physical Sciences Research Council grant [EP/N033922/1].

\subsection*{Notation} Let $k$ be a field of characteristic not two or three.
\vspace{0.2cm}

\noindent
For a finite-dimensional quadratic vector space $(V,q)$ and $i\in \mathbb{N}$ such that $i<char(k)$ if $0<char(k)$, we denote by $\eta : \Lambda^i(V)\rightarrow\Lambda^i(V)^*$ the canonical isomorphism given by the determinant and we consider the quadratic form $q_{\Lambda}$ (resp. $q_{\Lambda*}$) and the symmetric bilinear form $B_{\Lambda}$ associated by polarisation (resp. $B_{\Lambda*}$) on $\Lambda^i(V)$ (resp. $\Lambda^i(V)^*$) given by $\eta$.
\vspace{0.2cm}

\noindent
If $\lbrace e_i \rbrace$ is a basis of $V$, we denote $e_{i_1}\wedge\ldots\wedge e_{i_n}$ by $e_{i_1\ldots i_n}$.

\section{Lie superalgebras from special orthogonal representations}\label{section LSAs from special orthogonal rep}

In this section we explain how to construct a quadratic Lie superalgebra from an orthogonal representation of a quadratic Lie algebra, for details and proofs see \cite{Meyer19Kostant}. Let $(\mathfrak{g},B_{\mathfrak{g}})$ be a finite-dimensional quadratic Lie algebra, let $(V,(\phantom{v},\phantom{v}))$ be a finite-dimensional quadratic vector space and let $\rho : \mathfrak{g} \rightarrow \mathfrak{so}(V,(\phantom{v},\phantom{v}))$ be an orthogonal representation of $\mathfrak{g}$.
\vspace{0.1cm}

The moment map of the representation $\rho : \mathfrak{g} \rightarrow \mathfrak{so}(V,(\phantom{v},\phantom{v}))$ is the $\mathfrak{g}$-equivariant alternating map $\mu\in {\rm Alt}_2(V,\mathfrak{g})$ satisfying
$$B_{\mathfrak{g}}(x,\mu(v,w))=(\rho(x)(v),w) \qquad \forall x \in \mathfrak{g}, ~ \forall v,w \in V.$$
The standard example is the moment map of the fundamental representation of $\mathfrak{so}(V,(\phantom{v},\phantom{v}))$:

\begin{ex}
Suppose that $\mathfrak{g}=\mathfrak{so}(V,(\phantom{v},\phantom{v}))$ and $B_{\mathfrak{g}}(f,g)=-\frac{1}{2}Tr(fg)$ for all $f,g \in \mathfrak{so}(V,(\phantom{v},\phantom{v}))$. The corresponding moment map $\mu_{can}\in{\rm Alt}_2(V,\mathfrak{so}(V,(\phantom{v},\phantom{v})))$ satisfies
\begin{equation} \label{canonical moment map}
\mu_{can}(u,v)(w)=(u,w)v-(v,w)u \qquad \forall u,v,w\in V,
\end{equation}
and is a $\mathfrak{g}$-equivariant isomorphism between $\Lambda^2(V)$ and $\mathfrak{so}(V,(\phantom{v},\phantom{v}))$.
\end{ex}

We now define a particular class of orthogonal representations of quadratic Lie algebras:

\begin{df}
The representation $\rho : \mathfrak{g} \rightarrow \mathfrak{so}(V,(\phantom{v},\phantom{v}))$ is said to be special orthogonal if
\begin{equation}\label{eq CS}
    \mu(u,v)(w)+\mu(u,w)(v)=(u,v)w+(u,w)v-2(v,w)u \qquad \forall u,v,w\in V.
\end{equation}
\end{df}

Special orthogonal representations can be extended to define Lie superalgebras as follows:

\begin{thm}\label{thm LSA from special orthogonal rep}
Let $\rho : \mathfrak{g} \rightarrow \mathfrak{so}(V,(\phantom{v},\phantom{v}))$ be a finite-dimensional orthogonal representation of a finite-dimensional quadratic Lie algebra $(\mathfrak{g},B_{\mathfrak{g}})$ and let $\mathfrak{sl}(2,k)\rightarrow\mathfrak{sp}(k^2,\omega)$ be the symplectic fundamental representation of the quadratic Lie algebra $(\mathfrak{sl}(2,k),B_{\mathfrak{s}})$ where $\omega$ is the canonical symplectic form on $k^2$ and where $B_{\mathfrak{s}}(f,g)=\frac{1}{2}Tr(fg)$ for all $f,g\in \mathfrak{sl}(2,k)$. Let  $\tilde{\gg}$ be the super vector space defined by
$$\tilde{\gg}:=\mathfrak{g}\oplus\mathfrak{sl}(2,k)\oplus V\otimes k^2,$$
and let $B_{\tilde{\gg}}:=B_{\mathfrak{g}}\perp B_{\mathfrak{s}}\perp (\phantom{v},\phantom{v})\otimes \omega$. Then $(\tilde{\gg},B_{\tilde{\gg}},\lbrace \phantom{v},\phantom{v} \rbrace)$ is a quadratic Lie superalgebra extending the bracket of $\mathfrak{g}\oplus\mathfrak{sl}(2,k)$ and the action of $\mathfrak{g}\oplus\mathfrak{sl}(2,k)$ on $V\otimes k^2$ if and only if $\rho : \mathfrak{g} \rightarrow \mathfrak{so}(V,(\phantom{v},\phantom{v}))$ is a special orthogonal representation.
\end{thm}

In addition to the moment map, a trilinear and a quadrilinear alternating multilinear map can be naturally associated to a special orthogonal representation:

\begin{df} We define the multilinear alternating maps $\psi \in {\rm Alt}_3(V,V)$ and $Q\in {\rm Alt}_4(V,k)$ as follows:
\begin{align*}
\psi(v_1,v_2,v_3)&=\mu(v_1,v_2)(v_3)+\mu(v_3,v_1)(v_2)+\mu(v_2,v_3)(v_1), \\
Q(v_1,v_2,v_3,v_4)&=(v_1,\psi(v_2,v_3,v_4))-(v_4,\psi(v_1,v_2,v_3))+(v_3,\psi(v_4,v_1,v_2))-(v_2,\psi(v_3,v_4,v_1))
\end{align*}
for all $v_1,v_2,v_3,v_4 \in V$. The maps $\mu$, $\psi$ and $Q$ are called the covariants of $V$.
\end{df}

We have the following formulae:

\begin{pp}\label{pp formules lien BQ Bpsi dans le cas CS}
If $\rho : \mathfrak{g} \rightarrow \mathfrak{so}(V,(\phantom{v},\phantom{v}))$ is special orthogonal, then we have
\begin{align*}
\psi(v_1,v_2,v_3)&=3(\mu(v_1,v_2)(v_3)-\mu_{can}(v_1,v_2)(v_3)), \\
Q(v_1,v_2,v_3,v_4)&=4(v_1,\psi(v_2,v_3,v_4)),
\end{align*}
for all $v_1,v_2,v_3,v_4 \in V$.
\end{pp}

For vector spaces $E,F,G,H$, the exterior product $f\wedge_{\phi} g\in{\rm Alt}_{p+q}(E,H)$ of $f\in {\rm Alt}_p(E,F)$ and $g\in {\rm Alt}_q(E,G)$ relative to a bilinear map $\phi : F\times G\rightarrow H$ is defined by
$$f\wedge_{\phi} g(v_1,\ldots,v_{p+q})=\sum_{\sigma\in S(\llbracket 1,p \rrbracket,\llbracket p+1,p+q\rrbracket)}sgn(\sigma)\phi(f(v_{\sigma(1)},\ldots,v_{\sigma(p)}),g(v_{\sigma(p+1)},\ldots,v_{\sigma(p+q)}))$$
where the sum is over the $(p,q)$-shuffle permutations in $S_{p+q}$. If $\phi$ is implicit, then we denote $f\wedge_{\phi} g$ by $f\wedge g$. The composition $f\circ g\in{\rm Alt}_{pq}(G,F)$ of $f\in {\rm Alt}_p(E,F)$ and $g\in {\rm Alt}_q(G,E)$ is defined by
$$f\circ g(v_1,\ldots,v_{pq})=\sum_{\sigma\in S(\llbracket 1,q \rrbracket,\ldots,\llbracket p(q-1)+1,pq\rrbracket)}sgn(\sigma)f(g(v_{\sigma(1)},\ldots,v_{\sigma(q)}),\ldots,g(v_{\sigma(p(q-1)+1)},\ldots,v_{\sigma(pq)}))$$
where the sum is over the $(q,\ldots,q)$-shuffle permutations in $S_{pq}$.
\vspace{0.3cm}

Covariants of special orthogonal representations satisfy to the following Mathews identities:

\begin{thm} \label{thm id mathews}
Let $\rho : \mathfrak{g} \rightarrow \mathfrak{so}(V,(\phantom{v},\phantom{v}))$ be a finite-dimensional special orthogonal representation of a finite-dimensional quadratic Lie algebra and let $\mu\in {\rm Alt}_2(V,\mathfrak{g})$, $\psi\in {\rm Alt}_3(V,V)$ and $Q\in {\rm Alt}_4(V,k)$ be its covariants. We have the following identities:
\begin{alignat}{4}
a)&\qquad\qquad\qquad\qquad&\mu\wedge_{\rho} \psi&=-\frac{3}{2}Q\wedge Id_V \quad &&\in {\rm Alt}_5(V,V), \label{id premat} \\
b)&\qquad\qquad\qquad\qquad&\mu \circ \psi &= 3Q \wedge \mu \quad &&\in {\rm Alt}_6(V,\mathfrak{g}), \label{id mat1} \\
c)&\qquad\qquad\qquad\qquad&\psi\circ \psi&=-\frac{27}{2} Q\wedge Q\wedge Id_V \quad &&\in {\rm Alt}_9(V,V), \label{id mat2} \\
d)&\qquad\qquad\qquad\qquad& Q\circ \psi&=-54 Q\wedge Q \wedge Q \quad &&\in {\rm Alt}_{12}(V,k). \label{id mat3}
\end{alignat}
\end{thm}

\section{A one-parameter family of special orthogonal representations of $\mathfrak{sl}(2,k)\times\mathfrak{sl}(2,k)$}\label{section one-parameter family of special orthogonal representations}

In this section we show that with respect to a one parameter family of invariant quadratic forms on $\mathfrak{sl}(2,k)\times\mathfrak{sl}(2,k)$, the tensor product of the two fundamental representations is a special orthogonal representation.
\vspace{0.2cm}

Let $(V,\omega_V)$ and $(W,\omega_W)$ be two-dimensional symplectic vector spaces. The vector space $V\otimes W$ is quadratic for the symmetric bilinear form $\omega_V\otimes \omega_W$ given by
$$\omega_V\otimes \omega_W(v_1\otimes w_1,v_2\otimes w_2)=-\omega_V(v_1,v_2)\omega_W(w_1,w_2) \qquad \forall v_1,v_2\in V, ~ \forall w_1,w_2\in W.$$
Consider the bilinear form $K_V$ (resp. $K_W$) on $\mathfrak{sp}(V,\omega_V)$ (resp. $\mathfrak{sp}(W,\omega_W)$) defined by $K_V(f,g)=\frac{1}{2}Tr(fg)$ (resp. $K_W(f,g)=\frac{1}{2}Tr(fg)$) for all $f,g \in \mathfrak{sp}(V,\omega_V)$ (resp. $\mathfrak{sp}(W,\omega_W)$). For $\alpha,\beta\in k^*$, we now consider the orthogonal representation
$$\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W)\rightarrow \mathfrak{so}(V\otimes W, \omega_V\otimes \omega_W)$$
of the quadratic Lie algebra $(\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W),\frac{1}{\alpha}K_V\perp \frac{1}{\beta}K_W)$. Its moment map
$$\mu_{\alpha,\beta}:{\rm Alt}_2(V\otimes W,\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W))$$
satisfies
$$\mu_{\alpha,\beta}(v_1\otimes w_1,v_2\otimes w_2)=-\Big(\alpha \mu_V(v_1,v_2)\omega_W(w_1,w_2)+\beta \mu_W(w_1,w_2)\omega_V(v_1,v_2)\Big)\qquad \forall v_1,v_2\in V, ~ \forall w_1,w_2\in W,$$
where $\mu_i : S^2(V_i) \rightarrow \mathfrak{sp}(V_i,\omega_i)$ is the canonical symmetric moment map given by
$$\mu_i(v_1,v_2)(v_3)=-\omega_i(v_1,v_3)v_2-\omega_i(v_2,v_3)v_1 \qquad \forall v_1,v_2,v_3 \in V_i.$$

\begin{pp}
The orthogonal representation
$$\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W)\rightarrow \mathfrak{so}(V\otimes W, \omega_V\otimes \omega_W)$$
of the quadratic Lie algebra $(\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W),\frac{1}{\alpha}K_V\perp \frac{1}{\beta}K_W)$ is a special orthogonal representation if and only if $\alpha+\beta=-1$.
\end{pp}

\begin{proof}
Let $v_1\otimes w_1,v_2\otimes w_2, v_3\otimes w_3 \in V\otimes W$. We want to know under what conditions on $\alpha$ and $\beta$ do we have
\begin{align} \label{CS sl2xsl2}
&\mu_{\alpha,\beta}(v_1\otimes w_1,v_2\otimes w_2)(v_3\otimes w_3)+\mu_{\alpha,\beta}(v_1\otimes w_1,v_3\otimes w_3)(v_2\otimes w_2)=\omega_V\otimes \omega_W(v_1\otimes w_1,v_2\otimes w_2)v_3\otimes w_3 \nonumber \\
&+\omega_V\otimes \omega_W(v_1\otimes w_1,v_3\otimes w_3)v_2\otimes w_2-2\omega_V\otimes \omega_W(v_2\otimes w_2,v_3\otimes w_3)v_1\otimes w_1.
\end{align}
Since $V$ and $W$ are two-dimensional (and after a permutation of $v_1,v_2,v_3$ or $w_1,w_2,w_3$ if necessary) we have $v_3=av_1+bv_2$ and $w_3=cw_1+dw_2$ where $a,b,c,d \in k$. Hence we have
\begin{align*}
&\omega_V\otimes \omega_W(v_1\otimes w_1,v_2\otimes w_2)v_3\otimes w_3+\omega_V\otimes \omega_W(v_1\otimes w_1,v_3\otimes w_3)v_2\otimes w_2-2\omega_V\otimes \omega_W(v_2\otimes w_2,v_3\otimes w_3)v_1\otimes w_1 \\
&=\omega_V(v_1,v_2)\omega_W(w_1,w_2)\Big(-av_1\otimes dw_2-bv_2\otimes cw_1-2bv_2\otimes dw_2+av_1\otimes cw_1 \Big)
\end{align*}
On the other hand we have
\begin{align*}
&\mu_{\alpha,\beta}(v_1\otimes w_1,v_2\otimes w_2)(v_3\otimes w_3)+\mu_{\alpha,\beta}(v_1\otimes w_1,v_3\otimes w_3)(v_2\otimes w_2)\\
&=-\Big(\alpha\mu_V(v_1,v_2)(v_3)\otimes \omega_W(w_1,w_2)w_3+\beta\omega_V(v_1,v_2)v_3\otimes \mu_W(w_1,w_2)(w_3)+\alpha\mu_V(v_1,v_3)(v_2)\otimes \omega_W(w_1,w_3)w_2\\
&+\beta\omega_V(v_1,v_3)v_2\otimes \mu_W(w_1,w_3)(w_2)\Big)\\
&=(\alpha+\beta)\omega_V(v_1,v_3)v_2\otimes \omega_W(w_1,w_2)w_3+(\alpha+\beta)\omega_V(v_1,v_2)v_3\otimes \omega_W(w_1,w_3)w_2+\alpha\omega_V(v_2,v_3)v_1\otimes \omega_W(w_1,w_2)w_3\\
&+\alpha\omega_V(v_3,v_2)v_1\otimes \omega_W(w_1,w_3)w_2+\beta\omega_V(v_1,v_2)v_3\otimes \omega_W(w_2,w_3)w_1+\beta\omega_V(v_1,v_3)v_2\otimes \omega_W(w_3,w_2)w_1\\
&=(\alpha+\beta)\omega_V(v_1,v_2)bv_2\otimes \omega_W(w_1,w_2)cw_1+(\alpha+\beta)\omega_V(v_1,v_2)bv_2\otimes \omega_W(w_1,w_2)dw_2\\
&+(\alpha+\beta)\omega_V(v_1,v_2)av_1\otimes \omega_W(w_1,w_2)dw_2+(\alpha+\beta)\omega_V(v_1,v_2)bv_2\otimes \omega_W(w_1,w_2)dw_2\\
&+\alpha\omega_V(v_2,v_1)av_1\otimes \omega_W(w_1,w_2)cw_1+\alpha\omega_V(v_2,v_1)av_1\otimes \omega_W(w_1,w_2)dw_2+\alpha\omega_V(v_1,v_2)av_1\otimes \omega_W(w_1,w_2)dw_2\\
&+\beta\omega_V(v_1,v_2)av_1\otimes \omega_W(w_2,w_1)cw_1+\beta\omega_V(v_1,v_2)bv_2\otimes \omega_W(w_2,w_1)cw_1+\beta\omega_V(v_1,v_2)bv_2\otimes \omega_W(w_1,w_2)cw_1\\
&=(\alpha+\beta)\omega_V(v_1,v_2)\omega_W(w_1,w_2)\Big( bv_2\otimes cw_1+2bv_2\otimes dw_2+av_1\otimes dw_2-av_1\otimes cw_1\Big).
\end{align*}
Hence, Equation \eqref{CS sl2xsl2} is satisfied if and only if $\alpha+\beta=-1$ and so the representation $\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W)\rightarrow \mathfrak{so}(V\otimes W, \omega_V\otimes \omega_W)$ is special orthogonal if and only if $\alpha+\beta=-1$.
\end{proof}
\vspace{0.2cm}

Suppose that $\alpha+\beta=-1$. By the previous proposition and Theorem \ref{thm LSA from special orthogonal rep} we have a Lie superalgebra $\tilde{\gg}_{\alpha}$ of the form
$$\tilde{\gg}_{\alpha}=\mathfrak{sp}(V,\omega_V)\oplus \mathfrak{sp}(W,\omega_W)\oplus \mathfrak{sl}(2,k)\oplus V\otimes W \otimes k^2.$$
This a simple Lie superalgebra of type $D(2,1;\alpha)$ which is an exceptional simple Lie superalgebra if $\alpha$ is not equal to $-\frac{1}{2},-2$ or $1$.
\vspace{0.2cm}

\begin{rem}
\begin{enumerate}[label=\alph*)]
\item In \cite{Serganova1983}, Serganova shows that there are three families of simple real Lie superalgebras which are real forms of $D(2,1;\alpha)$ (see also \cite{Parker80} for a discussion about the real forms of $D(2,1;\alpha)$). If $k=\mathbb{R}$, the family $\tilde{\gg}_{\alpha}$ defined above corresponds to one these families.
\item There is a symmetry exchanging $\alpha$ and $\beta$. Hence, the special orthogonal representations $\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W)\rightarrow \mathfrak{so}(V\otimes W, \omega_V\otimes \omega_W)$ of the quadratic Lie algebras $(\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W),\frac{1}{\alpha}K_V\perp \frac{1}{-1-\alpha}K_W)$ and $(\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W),\frac{1}{-1-\alpha}K_V\perp \frac{1}{\alpha}K_W)$ give rise to isomorphic Lie superalgebras $\tilde{\gg}_{\alpha}$ and $\tilde{\gg}_{-1-\alpha}$.
\item There is a singular case when $\alpha=\beta=-\frac{1}{2}$. The Lie algebra $\mathfrak{sp}(V,\omega_V)\times\mathfrak{sp}(W,\omega_W)$ is isomorphic to $\mathfrak{so}(W_0,(\phantom{v},\phantom{v}))$, where $(W_0,(\phantom{v},\phantom{v}))$ is a four-dimensional hyperbolic vector space, and under this isomorphism, the quadratic form $\frac{1}{\alpha}K_V+\frac{1}{\alpha}K_W$ of $\mathfrak{sp}(V,\omega_V)\times\mathfrak{sp}(W,\omega_W)$ is isometric to the quadratic form $-\frac{1}{2}Tr(fg)$ for all $f,g \in \mathfrak{so}(W_0,(\phantom{v},\phantom{v}))$.  Hence, we have that $\tilde{\gg}_{-\frac{1}{2}}$ is isomorphic to $\mathfrak{osp}(W_0\oplus W_1,(\phantom{v},\phantom{v})\perp \omega)$ where $(W_1,\omega)$ is a two-dimensional symplectic vector space.
\end{enumerate}
\end{rem}

We now study the trilinear covariant and the quadrilinear covariant of the special orthogonal representation $\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W)\rightarrow \mathfrak{so}(V\otimes W, \omega_V\otimes \omega_W)$. Note that the Mathews identities of Theorem \ref{thm id mathews} vanish identically because $V\otimes W$ is of dimension four.

\begin{pp}
Suppose that $\alpha+\beta=-1$. The trilinear covariant $\psi\in {\rm Alt}_3(V\otimes W,V\otimes W)$ and the quadrilinear covariant $Q\in {\rm Alt}_4(V\otimes W,k)$ of the special orthogonal representation
$$\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W)\rightarrow \mathfrak{so}(V\otimes W, \omega_V\otimes \omega_W)$$
satisfies:
\begin{align*}
    \psi(v_1\otimes w_1,v_2\otimes w_2,v_3\otimes w_3)=&3(2\alpha+1)\Big( \omega_V(v_1,v_3)v_2\otimes \omega_W(w_3,w_2)w_1+\omega_V(v_2,v_3)v_1\otimes\omega_W(w_1,w_3)w_2\Big),\\
    Q(v_1\otimes w_1,v_2\otimes w_2,v_3\otimes w_3,v_4\otimes w_4)=&-12(2\alpha+1)\Big(\omega_V(v_2,v_4)\omega_V(v_1,v_3)\omega_W(w_4,w_3)\omega_W(w_1,w_2)\\
    &+\omega_V(v_3,v_4)\omega_V(v_1,v_2)\omega_W(w_2,w_4)\omega_W(w_1,w_3)\Big),
\end{align*}
for all $v_1,v_2,v_3,v_4\in V$, $w_1,w_2,w_3,w_4\in W$.
\end{pp}

\begin{proof}
Let $v_1\otimes w_1,v_2\otimes w_2,v_3\otimes w_3 \in V\otimes W$. Since $V$ and $W$ are two-dimensional (and after a permutation of $v_1,v_2,v_3$ or $w_1,w_2,w_3$ if necessary) we have $v_3=av_1+bv_2$ and $w_3=cw_1+dw_2$ where $a,b,c,d \in k$. We have
\begin{align}
\mu_{\alpha,\beta}(v_1\otimes w_1,v_2\otimes w_2)(v_3\otimes w_3)&=(\alpha+\beta)\omega_V(v_2,v_3)v_1\otimes\omega_W(w_3,w_2)w_1+(\alpha+\beta)\omega_V(v_1,v_3)v_2\otimes\omega_W(w_1,w_3)w_2\nonumber\\
&+(\alpha-\beta)\omega_V(v_2,v_3)v_1\otimes\omega_W(w_1,w_3)w_2+(\alpha-\beta)\omega_V(v_1,v_3)v_2\otimes\omega_W(w_3,w_2)w_1,\label{Bpsi D21 rel 1}\\
\mu_{\alpha,\beta}(v_2\otimes w_2,v_3\otimes w_3)(v_1\otimes w_1)&=(\alpha+\beta)\omega_V(v_2,v_3)v_1\otimes\omega_W(w_2,w_3)w_1+2\alpha\omega_V(v_3,v_1)v_2\otimes\omega_W(w_2,w_3)w_1\nonumber\\
&+2\beta\omega_V(v_2,v_3)v_1\otimes\omega_W(w_3,w_1)w_2,\label{Bpsi D21 rel 2}\\
\mu_{\alpha,\beta}(v_3\otimes w_3,v_1\otimes w_1)(v_2\otimes w_2)&=(\alpha+\beta)\omega_V(v_1,v_3)v_2\otimes\omega_W(w_3,w_1)w_2+2\alpha\omega_V(v_3,v_2)v_1\otimes\omega_W(w_3,w_1)w_2\nonumber\\
&+2\beta\omega_V(v_3,v_1)v_2\otimes\omega_W(w_3,w_2)w_1.\label{Bpsi D21 rel 3}
\end{align}
Hence, summing Equations \eqref{Bpsi D21 rel 1}, \eqref{Bpsi D21 rel 2} and \eqref{Bpsi D21 rel 3}, we obtain
$$\psi(v_1\otimes w_1,v_2\otimes w_2,v_3\otimes w_3)=3(\alpha-\beta)\Big( \omega_V(v_1,v_3)v_2\otimes \omega_W(w_3,w_2)w_1+\omega_V(v_2,v_3)v_1\otimes\omega_W(w_1,w_3)w_2\Big).$$
The formula for $Q$ follows by Proposition \ref{pp formules lien BQ Bpsi dans le cas CS}
\end{proof}

\begin{rem}
For the singular case $\alpha=-\frac{1}{2}$, we have that the covariants $\psi$ and $Q$ vanish identically. It means that the representation $\mathfrak{sp}(V,\omega_V)\times \mathfrak{sp}(W,\omega_W)\rightarrow \mathfrak{so}(V\otimes W, \omega_V\otimes \omega_W)$ is of $\mathbb{Z}_2$-Lie type in the sense of Kostant \cite{Kostant99} and then can be extended to define a Lie algebra structure on $\mathfrak{sp}(V,\omega_V)\oplus \mathfrak{sp}(W,\omega_W) \oplus V\otimes W$. This Lie algebra is isomorphic to the orthogonal Lie algebra $\mathfrak{so}(V\otimes W \oplus L,\omega_V\otimes \omega_W\perp (\phantom{u},\phantom{v})_L)$ where $(L,(\phantom{u},\phantom{v})_L)$ is a one-dimensional quadratic vector space.
\end{rem}

\section{The fundamental representation of $G_2$ is special orthogonal}\label{section G3}

In this section, we show that the irreducible $7$-dimensional fundamental representation of an exceptional Lie algebra $\mathfrak{g}$ of type $G_2$ is special orthogonal. To do this we realise $\mathfrak{g}$ as the derivation algebra of an octonion algebra $\mathbb{O}$ and use octonionic calculations. We first recall some properties of the octonions, for details and proofs see \cite{Schafer66} and \cite{Springer00}.
\vspace{0.2cm}

Let $\mathbb{O}$ be an octonion (or Cayley) algebra over $k$. This is a 8-dimensional unital composition algebra, the conjugation $\bar{\phantom{v}}$ satisfies $q(u)=u\bar{u}$ for all $u\in \mathbb{O}$, where $q$ is the norm of $\mathbb{O}$, and we have $\mathbb{O}=k\oplus {\rm Im}(\mathbb{O})$, where ${\rm Im}(\mathbb{O})=\lbrace u \in \mathbb{O} ~ | ~ \bar{u}=-u \rbrace$. Denote $B$ the symmetric bilinear form associated by polarisation to $q$. Let $e_1,e_2,e_4 \in {\rm Im}(\mathbb{O})$ be such that $\mathcal{B}=\lbrace e_1,e_2,e_1e_2,e_4,e_1e_4,e_2e_4,(e_1e_2)e_4 \rbrace$ is an orthogonal and anisotropic basis of ${\rm Im}(\mathbb{O})$ and set $e_3:=e_1e_2$, $e_5:=e_1e_4$, $e_6:=e_2e_4$, $e_7:=(e_1e_2)e_4$. This basis is related to the Fano plane:
\begin{center}
\begin{tikzpicture}
\draw (0,0) -- (4,0);
\draw (0,0) -- (2,3.46410161514);
\draw (2,3.46410161514) -- (4,0);
\draw (2,1.15470053838) circle (1.15470053838) ;
\draw (2,3.46410161514) -- (2,0);
\draw (0,0) -- (3,1.73205080757);
\draw (4,0) -- (1,1.73205080757);
\draw (0,0) node[below]{$6$};
\draw (2,0) node[below]{$3$};
\draw (4,0) node[below]{$5$};
\draw (1,1.73205080757) node[left]{$1$};
\draw (3,1.73205080757) node[right]{$2$};
\draw (2,3.46410161514) node[above]{$7$};
\draw (2.2,1.15470053838) node[right]{$4$};
\end{tikzpicture}
\end{center}
in the sense that, for $i\neq j$, the product between $e_i$ and $e_j$ is a multiple of $e_k$ where $k$ is the third point on the line going through $i$ and $j$.
\vspace{0.2cm}

The commutator and the associator are the alternating maps given by:
\begin{align*}
[u,v]&=uv-vu,\\
(u,v,w)&=(uv)w-u(vw)
\end{align*}
for all $u,v,w\in \mathbb{O}$. The commutator doesn't define a Lie algebra structure on $\mathbb{O}$ since the Jacobi tensor $J$ satisfies
\begin{equation}\label{eq Jacobi G2}
J(u,v,w)=[u,[v,w]]+[v,[w,u]]+[w,[u,v]]=-6(u,v,w) \qquad \forall u,v,w\in \mathbb{O}.
\end{equation}
There is a cross-product on $\mathbb{O}$ defined by
$$u\times v=\frac{1}{2}(\bar{v}u-\bar{u}v)\qquad \forall u,v\in\mathbb{O},$$
and we have
\begin{alignat}{2}
q(u\times v)&=q_{\Lambda}(u\wedge v)=q(u)q(v)-B(u,v)^2 \qquad &&\forall u,v\in \mathbb{O},\label{eq length cross product}\\
u\times v&=\frac{1}{2}[u,v]=uv+B(u,v) \qquad &&\forall u,v,w\in {\rm Im}(\mathbb{O}),\label{eq link product and cross product}\\
u\times(v\times w)+v\times(u\times w)&=B(v,w)u+B(u,w)v-2B(u,v)w \qquad &&\forall u,v,w\in {\rm Im}(\mathbb{O}).\label{eq Malcev id}
\end{alignat}
The associative form $\phi$ on ${\rm Im}(\mathbb{O})$ is the trilinear alternating form defined by
$$\phi(u,v,w)=B(u,v\times w) \qquad \forall u,v,w\in {\rm Im}(\mathbb{O}),$$
and we have
\begin{align}
    \eta^{-1}(\phi)&=\frac{1}{q(e_1)q(e_2)}e_{123}-\frac{1}{q(e_1)q(e_2)q(e_4)}e_{167}+\frac{1}{q(e_1)q(e_2)q(e_4)}e_{257}-\frac{1}{q(e_1)q(e_2)q(e_4)}e_{356}\nonumber\\
    &+\frac{1}{q(e_1)q(e_4)}e_{145}+\frac{1}{q(e_2)q(e_4)}e_{246}+\frac{1}{q(e_1)q(e_2)q(e_4)}e_{347}.\label{decomp phi G2}
\end{align}

\begin{comment}
\begin{proof}
\begin{align*}
    u\times(v\times w)+v\times(u\times w)&=u(vw)+B(u,vw)+B(v,w)u+v(uw)+B(v,uw)+vB(u,w)\\
    &=-2B(u,v)w+B(u,vw)+B(v,w)u+B(v,uw)+vB(u,w)
\end{align*}
and since
$$B(u,vw)+B(v,uw)=0$$
then we obtain the equation.
\end{proof}
\end{comment}
\vspace{0.2cm}

Let $\rho : {\rm Im}(\mathbb{O}) \rightarrow {\rm End}(\mathbb{O})$ be the map defined by $\rho(u)(x)=ux$ for $u \in {\rm Im}(\mathbb{O})$ and $x\in \mathbb{O}$. We have
$$\rho(u)^2=-q(u)Id \qquad \forall u\in {\rm Im}(\mathbb{O})$$
and so $\rho$ extends to the Clifford algebra $C({\rm Im}(\mathbb{O}),-q)$. The quantisation map $Q:\Lambda({\rm Im}(\mathbb{O}))\rightarrow C({\rm Im}(\mathbb{O}),-q)$ is an $O({\rm Im}(\mathbb{O}),B)$-equivariant isomorphism of vector spaces and then we have $C({\rm Im}(\mathbb{O}),-q)=\bigoplus\limits_i C^i({\rm Im}(\mathbb{O}),-q)$ where $C^i({\rm Im}(\mathbb{O}),-q)=Q(\Lambda^i({\rm Im}(\mathbb{O})))$. The map $\mu_{can}\circ Q^{-1} : C^2({\rm Im}(\mathbb{O}),-q) \rightarrow \mathfrak{so}({\rm Im}(\mathbb{O}),q)$ is an isomorphism of Lie algebras. Let
$$\mathfrak{g}:=\lbrace x\in C^2({\rm Im}(\mathbb{O}),-q) ~ | ~ \rho(x)(1)=0 \rbrace.$$
This is a Lie algebra of type $G_2$, the map $\rho : \mathfrak{g}\rightarrow \mathfrak{so}({\rm Im}(\mathbb{O}),q)$ is its $7$-dimensional fundamental representation and $\rho(\mathfrak{g})$ is equal to the set of derivations of $\mathbb{O}$. Define the ad-invariant quadratic form $B_{\mathfrak{g}}$ on $\mathfrak{g}$ by
$$B_{\mathfrak{g}}(x,y)=-\frac{1}{3} Tr(\rho(x)\rho(y))\qquad\forall x,y\in \mathfrak{g}.$$

\begin{pp}\label{pp mu G2}
The moment map $\mu_{{\rm Im}}: \Lambda^2({\rm Im}(\mathbb{O}))\rightarrow \mathfrak{g}$ satisfies
$$\mu_{{\rm Im}}(u,v)(w)=-\frac{1}{4}([w,[u,v]]+3(u,v,w))\qquad\forall u,v,w\in {\rm Im}(\mathbb{O})$$
\end{pp}

\begin{proof}
For $u,v\in {\rm Im}(\mathbb{O})$, let $D(u,v)\in \mathfrak{g}$ be such that $\rho(D(u,v))(x)=[x,[u,v]]+3(u,v,x)$ for all $x\in {\rm Im}(\mathbb{O})$. Let $D$ in $\mathfrak{g}$. We want to show that
$$Tr(\rho(D)\rho(D(u,v))=12B(D(u),v).$$
Without loss of generality (changing $\mathcal{B}$ if necessary) we can assume that $u=e_1$ and $v=e_2$.
First of all
$$Tr(\rho(D)\rho(D(u,v))=\sum\limits_{e_i \in \mathcal{B}}\frac{1}{q(e_i)}B(\rho(D)(\rho(D(u,v))(e_i)),e_i).$$
We have
\begin{align*}
    \frac{1}{q(e_1)}B(D(D(e_1,e_2)(e_1)),e_1)&=4B(D(e_1),e_2),\\
    \frac{1}{q(e_1)}B(D(D(e_1,e_2)(e_2)),e_2)&=4B(D(e_1),e_2),\\
    \frac{1}{q(e_1e_2)}B(D(D(e_1,e_2)(e_1e_2)),e_1e_2)&=0,\\
    \frac{1}{q(e_3)}B(D(D(e_1,e_2)(e_3)),e_3)&=-\frac{2}{q(e_3)}B(D(e_3),(e_1e_2)e_3),\\
    \frac{1}{q(e_1e_3)}B(D(D(e_1,e_2)(e_1e_3)),e_1e_3)&=2B(D(e_1),e_2)+\frac{2}{q(e_3)}B(D(e_3),(e_1e_2)e_3),\\
    \frac{1}{q(e_2e_3)}B(D(D(e_1,e_2)(e_2e_3)),e_2e_3)&=2B(D(e_1),e_2)+\frac{2}{q(e_3)}B(D(e_3),(e_1e_2)e_3),\\
    \frac{1}{q((e_1e_2)e_3)}B(D(D(e_1,e_2)((e_1e_2)e_3)),(e_1e_2)e_3)&=-\frac{2}{q(e_3)}B(D(e_3),(e_1e_2)e_3),
\end{align*}
\begin{comment}
$$D(u,v)(u)=-4q(u)v$$
$$D(u,v)(v)=4q(v)u$$
$$D(u,v)(uv)=0$$
$$D(u,v)(w)=2(uv)w$$
$$D(u,v)(uw)=-2q(u)vw$$
$$D(u,v)(vw)=2q(v)uw$$
$$D(u,v)((uv)w)=-2q(u)q(v)w$$
\end{comment}

and hence
$$Tr(\rho(D)\rho(D(u,v)))=12B(D(u),v).$$
\end{proof}

\begin{cor} \label{expression mu g2 avec mucan}
For $u,v,w\in {\rm Im}(\mathbb{O})$, we have
\begin{enumerate}[label=\alph*)]
\item $\mu_{\rm Im}(u,v\times w)+\mu_{\rm Im}(w,u\times v)+\mu_{\rm Im}(v,w\times u)=0,$
\item $\mu_{{\rm Im}}(u,v)(w)=\frac{3}{2}\mu_{can}(u,v)(w)+\frac{1}{8}[w,[u,v]].$
\end{enumerate}
\end{cor}
\vspace{0.1cm}

\begin{proof}
$a)$ See (3.73) p.78 of \cite{Schafer66}.
\vspace{0.2cm}

\noindent
$b)$ Let $u,v,w\in {\rm Im}(\mathbb{O})$. We first show that
\begin{equation}\label{proof expression mu g2 avec mucan}
    -\frac{1}{4}[w,[u,v]]-\frac{1}{2}(u,v,w)=\mu_{can}(u,v)(w).
\end{equation}
Suppose that $u$ and $v$ are anisotropic and orthogonal.
\vspace{0.2cm}

\noindent
$\bullet$ If $w=u$ then \eqref{proof expression mu g2 avec mucan} follows from
$$-\frac{1}{4}[w,[u,v]]-\frac{1}{2}(u,v,w)=-\frac{1}{2}[u,uv]=-u^2v=q(u)v=\mu_{can}(u,v)(u).$$

\noindent
$\bullet$ If $w=uv$ then \eqref{proof expression mu g2 avec mucan} is clear since $[w,[u,v]]=(u,v,w)=\mu_{can}(u,v)(w)=0$.
\vspace{0.2cm}

\noindent
$\bullet$ If $\lbrace u,v,uv,w \rbrace$ are orthogonal then we have $\mu_{can}(u,v)(w)=0$ and
$$-\frac{1}{4}[w,[u,v]]=-w(uv)=(uv)w=\frac{1}{2}(u,v,w).$$
Hence \eqref{proof expression mu g2 avec mucan} is satisfied and this proves the corollary using Proposition \ref{pp mu G2}.
\end{proof}
\vspace{0.1cm}

We now give the main result of this section.
\vspace{0.1cm}

\begin{thm} \label{pp ImO is special}
The representation $\rho : \mathfrak{g} \rightarrow \mathfrak{so}({\rm Im}(\mathbb{O}),B)$ of the quadratic Lie algebra $(\mathfrak{g},B_{\mathfrak{g}})$ is a special orthogonal representation.
\end{thm}
\begin{proof}
Let $u,v,w\in {\rm Im}(\mathbb{O})$. Using Proposition \ref{pp mu G2} and \eqref{eq Malcev id} we have
\begin{align*}
    \mu_{\rm Im}(u,v)(w)+\mu_{\rm Im}(u,w)(v)&=-\frac{1}{4}([w,[u,v]]+[v,[u,w]])=-w\times (u\times v)-v\times(u\times w)\\
    &=w\times (v\times u)+v\times (w\times u)=B(u,v)w+B(u,w)v-2B(w,v)u.
\end{align*}
\end{proof}

By Theorems \ref{thm LSA from special orthogonal rep} and \ref{pp ImO is special} we have a Lie superalgebra $\tilde{\gg}$ of the form
$$\tilde{\gg}=\mathfrak{g}\oplus \mathfrak{sl}(2,k)\oplus {\rm Im}(\mathbb{O}) \otimes k^2.$$
This is an exceptional simple Lie superalgebra of type $G_3$.

\begin{rem}
If $k=\mathbb{R}$, Serganova (see \cite{Serganova1983}) showed that there are two real forms of $G_3$ whose even parts are isomorphic to the compact (resp. split) exceptional simple real Lie algebra of type $G_2$ in direct sum with $\mathfrak{sl}(2,\mathbb{R})$ and whose odd parts are isomorphic to the tensor product of the fundamental representations. In our construction, if $\mathbb{O}$ is the compact (resp. split) octonion algebra, the Lie algebra $\mathfrak{g}$ is the compact (resp. split) exceptional simple real Lie algebra of type $G_2$ and both real forms of $G_3$ are obtained by our construction.
\end{rem}

Since the representation $\mathfrak{g} \rightarrow \mathfrak{so}({\rm Im}(\mathbb{O}),B)$ is special, we calculate its covariants and the Mathews identities they satisfy. Both-sides of Equations \eqref{id mat2} and \eqref{id mat3} vanish identically since ${\rm Im}(\mathbb{O})$ is of dimension $7$. It turns out, that both-sides of Equation \eqref{id mat1} also vanish identically. However, both sides of Equation \eqref{id premat} do not vanish identically and, up to constants, $Q_{\rm Im}\wedge Id$ and $\mu_{\rm Im}\wedge_{\rho} \psi_{\rm Im}\in {\rm Alt}_5({\rm Im}(\mathbb{O}),{\rm Im}(\mathbb{O}))$ are the Hodge duals of the cross-product $\times \in {\rm Alt}_2({\rm Im}(\mathbb{O}),{\rm Im}(\mathbb{O}))$.

\begin{pp}\label{pp covariants ImO}
Let $\mu_{\rm Im},\psi_{\rm Im},Q_{\rm Im}$ be the covariants of the special orthogonal representation $\rho : \mathfrak{g} \rightarrow \mathfrak{so}({\rm Im}(\mathbb{O}),B)$. We have
\begin{enumerate}[label=\alph*)]
\item $\psi_{\rm Im}(v_1,v_2,v_3)=-\frac{3}{4}(v_1,v_2,v_3)~$ for all $v_1,v_2,v_3\in {\rm Im}(\mathbb{O})$,
\item $Q_{\rm Im}(v_1,v_2,v_3,v_4)=-3B(v_1,(v_2,v_3,v_4))~$ for all $v_1,v_2,v_3,v_4\in {\rm Im}(\mathbb{O})$,
\item  \begin{align}
        \eta^{-1}(Q_{\rm Im})&=\frac{6}{q(e_1)q(e_2)q(e_4)}e_{1247}-\frac{6}{q(e_1)q(e_2)q(e_4)}e_{1256}-\frac{6}{q(e_1)q(e_2)q(e_4)}e_{1346}-\frac{6}{q(e_1)^2q(e_2)q(e_4)}e_{1357}\nonumber\\
        &+\frac{6}{q(e_1)q(e_2)q(e_4)}e_{2345}-\frac{6}{q(e_1)q(e_2)^2q(e_4)}e_{2367}-\frac{6}{q(e_1)q(e_2)q(e_4)^2}e_{4567},\label{decomp Q G2}
        \end{align}
\item $\mu_{\rm Im} \circ \psi_{\rm Im}=0$ and $Q_{\rm Im}\wedge \mu_{\rm Im}=0$.
\end{enumerate}
\end{pp}
\begin{proof}
$a)$ By Proposition \ref{pp mu G2} and by Equation \eqref{eq Jacobi G2} we obtain
\begin{align*}
    \psi_{\rm Im}(v_1,v_2,v_3)&=-\frac{1}{4}(J(v_1,v_2,v_3)+3(v_1,v_2,v_3)+3(v_2,v_3,v_1)+3(v_3,v_1,v_2))\\
    &=-\frac{1}{4}(J(v_1,v_2,v_3)+9(v_1,v_2,v_3))\\
    &=-\frac{3}{4}(v_1,v_2,v_3).
\end{align*}

\noindent
$b)$ Follows from Proposition \ref{pp formules lien BQ Bpsi dans le cas CS}.
\vspace{0.2cm}

\noindent
$c)$ The decomposition follows from $b)$ and the fact that for $i_1<i_2<i_3<i_4$, then $Q_{\rm Im}(e_{i_1},e_{i_2},e_{i_3},e_{i_4})$ is non-zero if and only if $(i_1,i_2,i_3,i_4)\in \lbrace (1,2,4,7),(1,2,5,6),(1,3,4,6),(1,3,5,7),(2,3,4,5),(2,3,6,7),(4,5,6,7)\rbrace$.
\vspace{0.2cm}

\noindent
$d)$ Let $v_1,\ldots,v_6 \in {\rm Im}(\mathbb{O})$. We have
\begin{align*}
\mu_{\rm Im} \circ \psi_{\rm Im} (v_1,\ldots,v_6)&=\sum \limits_{\sigma \in S(\llbracket1,3\rrbracket,\llbracket 4,6 \rrbracket)}sgn(\sigma) \mu_{\rm Im} (\psi_{\rm Im}(v_{\sigma(1)},v_{\sigma(2)},v_{\sigma(3)}),\psi_{\rm Im}(v_{\sigma(4)},v_{\sigma(5)},v_{\sigma(6)})) \\
&=2\sum \limits_{\sigma \in S'}sgn(\sigma) \mu_{\rm Im} (\psi_{\rm Im}(v_{\sigma(1)},v_{\sigma(2)},v_{\sigma(3)}),\psi_{\rm Im}(v_{\sigma(4)},v_{\sigma(5)},v_{\sigma(6)}))\\
&=\frac{9}{8}\sum \limits_{\sigma \in S'}sgn(\sigma) \mu_{\rm Im} ((v_{\sigma(1)},v_{\sigma(2)},v_{\sigma(3)}),(v_{\sigma(4)},v_{\sigma(5)},v_{\sigma(6)}))
\end{align*}
where $S':=\lbrace Id,(14),(15),(16),(24),(25),(26),(34),(35),(36) \rbrace$.
\vspace{0.2cm}

Suppose that $v_i\in \mathcal{B}$ for all $i\in \llbracket 1,6 \rrbracket$. Since there is no distinguished way to choose 5 different points on the Fano plane, then, without loss of generality, we can assume that $v_i=e_i$ for all $i\in \llbracket 1,6 \rrbracket$. Since
$$(e_1,e_2,e_3)=(e_1,e_4,e_5)=(e_2,e_4,e_6)=(e_3,e_5,e_6)=0$$
then we have
\begin{align*}
    \mu_{\rm Im} \circ \psi_{\rm Im} (v_1,\ldots,v_6)&=-\frac{9}{8}\sum \limits_{\sigma \in S''}\mu_{\rm Im} ((e_{\sigma(1)},e_{\sigma(2)},e_{\sigma(3)}),(e_{\sigma(4)},e_{\sigma(5)},e_{\sigma(6)}))\\
    &=-\frac{9}{2}\sum \limits_{\sigma \in S''}\mu_{\rm Im} ((e_{\sigma(1)}e_{\sigma(2)})e_{\sigma(3)},(e_{\sigma(4)}e_{\sigma(5)})e_{\sigma(6)})
\end{align*}
where $S'':=\lbrace (14),(15),(24),(26),(35),(36) \rbrace$. Hence, we have that
$$\mu_{\rm Im} \circ \psi_{\rm Im}(v_1,\ldots,v_6)=-9q(e_1)q(e_2)q(e_4)\Big(\mu_{\rm Im} (e_1 e_2,e_4)+\mu_{\rm Im} (e_2 e_4,e_1)+\mu_{\rm Im} (e_4 e_1,e_2)\Big)$$
and so, using $a)$ of Corollary \ref{expression mu g2 avec mucan}, we obtain $\mu_{\rm Im} \circ \psi_{\rm Im}=0$ and by Theorem \ref{thm id mathews} we have $Q_{\rm Im}\wedge \mu_{\rm Im} =0$.
\end{proof}
\vspace{0.1cm}

\begin{rem}
\begin{enumerate}[label=\alph*)]
    \item We have
    $$\phi\wedge Q_{\rm Im}(e_1\wedge\ldots\wedge e_7)=-42q(e_1)^2q(e_2)^2q(e_4)^2.$$

    If $char(k)\neq 7$, then $\phi\wedge Q_{\rm Im}$ defines an orientation on ${\rm Im}(\mathbb{O})$
    \item In the decomposition \eqref{decomp phi G2} (resp. \eqref{decomp Q G2}), the seven quadruples of indices $\lbrace i_1,i_2,i_3,i_4\rbrace$ appearing are exactly (resp. the complements of) the seven lines of the Fano plane.
\end{enumerate}
\end{rem}

Suppose that $char(k)=0$ or $char(k)>7$. Define a quadratic form $B_{{\rm Alt}}$ on ${\rm Alt}_i({\rm Im}(\mathbb{O}),{\rm Im}(\mathbb{O}))\cong\Lambda^i({\rm Im}(\mathbb{O}))^*\otimes {\rm Im}(\mathbb{O})$ to be the tensor product of $B_{\Lambda^*}$ and $B$. For $f\in {\rm Alt}_i({\rm Im}(\mathbb{O}),{\rm Im}(\mathbb{O}))$ define its Hodge dual $*f\in {\rm Alt}_{7-i}({\rm Im}(\mathbb{O}),{\rm Im}(\mathbb{O}))$ to be the unique element which satisfies
$$\alpha\wedge_{B} *f=B_{{\rm Alt}}(\alpha,f)\phi\wedge Q_{\rm Im} \qquad \forall \alpha \in {\rm Alt}_i({\rm Im}(\mathbb{O}),{\rm Im}(\mathbb{O})).$$

\begin{pp}
We have
$$*\times=\frac{147}{8}Q_{\rm Im}\wedge Id=-\frac{49}{4}\mu_{\rm Im}\wedge_{\rho} \psi_{\rm Im}.$$
\end{pp}

\begin{proof}
Let $\alpha\in {\rm Alt}_2({\rm Im}(\mathbb{O}),{\rm Im}(\mathbb{O}))$. We have
\begin{equation}\label{eq proof hodge cross product Im O 1}
B_{{\rm Alt}}(\alpha,\times)\phi\wedge Q_{\rm Im}(e_1\wedge \ldots \wedge e_7)=-42\sum\limits_{i<j}\frac{1}{q(e_i)q(e_j)}B(\alpha(e_i,e_j),e_i\times e_j)q(e_1)^2q(e_2)^2q(e_4)^2.
\end{equation}
On the other hand
\begin{align*}
    \alpha\wedge_B Q_{\rm Im}\wedge Id(e_1\wedge\ldots\wedge e_7)&=\frac{1}{126}\sum\limits_{\sigma\in S_7}sgn(\sigma)B(\alpha(e_{\sigma(1)},e_{\sigma(2)}),e_{\sigma(3)})Q_{\rm Im}(e_{\sigma(4)},e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)}).\\
\end{align*}
Since $\alpha$ and $Q_{\rm Im}$ are alternating and using the decomposition of Equation \eqref{decomp Q G2}, we have
\begin{equation}\label{eq proof hodge cross product Im O 2}
    \alpha\wedge_B Q_{\rm Im}\wedge Id(e_1\wedge\ldots\wedge e_7)=\frac{8}{21}\sum\limits_{\sigma\in S}sgn(\sigma)B(\alpha(e_{\sigma(1)},e_{\sigma(2)}),e_{\sigma(3)})Q_{\rm Im}(e_{\sigma(4)},e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)})
\end{equation}
where
$$S=\lbrace\sigma\in S_7 ~ | ~ \sigma(1)<\sigma(2), \quad (\sigma(4),\sigma(5),\sigma(6),\sigma(7))\in \lbrace (1,2,4,7),\ldots,(4,5,6,7) \rbrace\rbrace.$$
We have $|S|=21$, each summand in \eqref{eq proof hodge cross product Im O 2} correspond to one summand in \eqref{eq proof hodge cross product Im O 1} and so, a straightforward calculation gives
$$\alpha\wedge_B Q_{\rm Im}\wedge Id(e_1\wedge\ldots\wedge e_7)=-\frac{16}{7}\sum\limits_{i<j}\frac{1}{q(e_i)q(e_j)}B(\alpha(e_i,e_j),e_i\times e_j)q(e_1)^2q(e_2)^2q(e_4)^2$$
and hence
$$\alpha\wedge_B Q_{\rm Im}\wedge Id=\frac{8}{147}B_{\rm Alt}(\alpha,\times)\phi \wedge Q_{\rm Im}.$$
\end{proof}

\begin{rem}
One can show similarly that, up to constants,
the identity is the Hodge dual of $\phi\wedge \psi_{\rm Im}$, the covariant $\mu_{\rm Im}$ is the Hodge dual of $\phi\wedge \mu_{\rm Im}$ and the covariant $\psi_{\rm Im}$ is the Hodge dual of $\phi\wedge Id$.
\end{rem}

\begin{comment}
\begin{pp}
We have
\begin{enumerate}[label=\alph*)]
\item $*\mu_{\rm Im}=-42\phi\wedge \mu_{\rm Im}$,
\item $*\psi_{\rm Im}=378\phi\wedge Id$,
\item $*\times=\frac{147}{8}Q_{\rm Im}\wedge Id=-\frac{49}{4}\mu_{\rm Im}\wedge_{\rho} \psi_{\rm Im}$.
\end{enumerate}
\end{pp}

\begin{proof}
$a)$ We have
$$B(\alpha,\mu_{\rm Im})\phi\wedge Q_{\rm Im}(e_1\wedge\ldots\wedge e_7)=-42\sum\limits_{i<j}\frac{1}{q(e_i)q(e_j)}B_{\mathfrak{g}}(\alpha(e_i,e_j),\mu_{\rm Im}(e_i,e_j))q(e_1)^2q(e_2)^2q(e_4)^2.$$
On the other hand
$$\alpha\wedge \phi\wedge\mu_{\rm Im}(e_1\wedge\ldots\wedge e_7)=\frac{1}{24}\sum\limits_{\sigma \in S_7}sgn(\sigma)B_{\mathfrak{g}}(\alpha(e_{\sigma(1)},e_{\sigma(2)}),\mu_{\rm Im}(e_{\sigma(3)},e_{\sigma(4)}))\phi(e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)})$$
$$\alpha\wedge \phi\wedge\mu_{\rm Im}(e_1\wedge\ldots\wedge e_7)=\sum\limits_{\sigma \in S}sgn(\sigma)B_{\mathfrak{g}}(\alpha(e_{\sigma(1)},e_{\sigma(2)}),\mu_{\rm Im}(e_{\sigma(3)},e_{\sigma(4)}))\phi(e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)}),$$
where $S=\lbrace \sigma \in S_7 ~ | ~ \sigma(1)<\sigma(2), ~ \sigma(3)<\sigma(4), ~ (\sigma(5),\sigma(6),\sigma(7))\in \lbrace (1,2,3), \ldots, \ldots (3,4,7) \rbrace \rbrace$
$$\alpha\wedge \phi\wedge\mu_{\rm Im}(e_1\wedge\ldots\wedge e_7)=\sum\limits_{i<j}\frac{1}{q(e_i)q(e_j)}B_{\mathfrak{g}}(\alpha(e_i,e_j),\mu_{\rm Im}(e_i,e_j))q(e_1)^2q(e_2)^2q(e_4)^2,$$
and so
$$\alpha\wedge\phi\wedge \mu_{\rm Im}=-\frac{1}{42}B(\alpha,\mu_{\rm Im})\phi\wedge Q_{\rm Im}.$$
\vspace{0.2cm}

\noindent
$b)$ We have
$$B(\alpha,\psi_{\rm Im})\phi\wedge Q_{\rm Im}(e_1\wedge\ldots\wedge e_7)=-42\sum\limits_{i<j<k}\frac{1}{q(e_i)q(e_j)q(e_k)}B_{\mathfrak{g}}(\alpha(e_i,e_j,e_k),\psi_{\rm Im}(e_i,e_j,e_k))q(e_1)^2q(e_2)^2q(e_4)^2.$$
On the other hand
$$\alpha\wedge \phi \wedge Id(e_1\wedge\ldots \wedge e_7)=-\frac{1}{36}\sum\limits_{\sigma\in S_7}sgn(\sigma)B(\alpha(e_{\sigma(1)},e_{\sigma(2)},e_{\sigma(3)}),e_{\sigma(4)})\phi(e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)})$$
$$\alpha\wedge \phi \wedge Id(e_1\wedge\ldots \wedge e_7)=-\frac{1}{6}\sum\limits_{\sigma\in S}sgn(\sigma)B(\alpha(e_{\sigma(1)},e_{\sigma(2)},e_{\sigma(3)}),e_{\sigma(4)})\phi(e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)})$$
where $S=\lbrace \sigma \in S_7 ~ | ~ \sigma(1)<\sigma(2),<\sigma(3), ~ (\sigma(5),\sigma(6),\sigma(7))\in \lbrace (1,2,3),\ldots \rbrace~ \rbrace$

and so
$$\alpha\wedge \phi \wedge Id=\frac{1}{6*21*3}B(\alpha,\phi_{\rm Im})\phi\wedge Q_{\rm Im}.$$
\vspace{0.2cm}

\noindent
$c)$ Let $\alpha\in {\rm Alt}_2({\rm Im}(\mathbb{O}),{\rm Im}(\mathbb{O}))$. We have
$$B(\alpha,\beta)=\sum\limits_{i<j}\frac{1}{q(e_i)q(e_j)}B(\alpha(e_i,e_j),\beta(e_i,e_j)) \qquad \forall \beta \in {\rm Alt}_2({\rm Im}(\mathbb{O}),{\rm Im}(\mathbb{O}))$$
and so
\begin{equation}\label{eq proof hodge cross product Im O 1}
B(\alpha,\times)\phi\wedge Q_{\rm Im}(e_1\wedge \ldots \wedge e_7)=-42\sum\limits_{i<j}\frac{1}{q(e_i)q(e_j)}B(\alpha(e_i,e_j),e_i\times e_j)q(e_1)^2q(e_2)^2q(e_4)^2.
\end{equation}
On the other hand
\begin{align*}
    \alpha\wedge Q_{\rm Im}\wedge Id(e_1\wedge\ldots\wedge e_7)&=\frac{1}{126}\sum\limits_{\sigma\in S_7}sgn(\sigma)B(\alpha(e_{\sigma(1)},e_{\sigma(2)}),e_{\sigma(3)})Q_{\rm Im}(e_{\sigma(4)},e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)}).\\
\end{align*}
Since $\alpha$ and $Q_{\rm Im}$ are alternating and using the decomposition of Equation \eqref{decomp Q G2}, we have
\begin{equation}\label{eq proof hodge cross product Im O 2}
    \alpha\wedge Q_{\rm Im}\wedge Id(e_1\wedge\ldots\wedge e_7)=\frac{2\times 4!}{126}\sum\limits_{\sigma\in S}sgn(\sigma)B(\alpha(e_{\sigma(1)},e_{\sigma(2)}),e_{\sigma(3)})Q_{\rm Im}(e_{\sigma(4)},e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)})
\end{equation}
where
$$S=\lbrace\sigma\in S_7 ~ | ~ \sigma(1)<\sigma(2), \quad (\sigma(4),\sigma(5),\sigma(6),\sigma(7))\in \lbrace (1,2,4,7),\ldots,(4,5,6,7) \rbrace\rbrace.$$
We have $|S|=21$, each summand in \eqref{eq proof hodge cross product Im O 2} correspond to one summand in \eqref{eq proof hodge cross product Im O 1} and so, a straightforward calculation gives
$$\alpha\wedge Q_{\rm Im}\wedge Id(e_1\wedge\ldots\wedge e_7)=-\frac{2\times 4!\times 6}{126}\sum\limits_{i<j}\frac{1}{q(e_i)q(e_j)}B(\alpha(e_i,e_j),e_i\times e_j)q(e_1)^2q(e_2)^2q(e_4)^2$$
and hence
$$\alpha\wedge Q_{\rm Im}\wedge Id=\frac{8}{147}B(\alpha,\times)\phi \wedge Q_{\rm Im}.$$
\end{proof}
\end{comment}

\section{The spinor representation of a Lie algebra of type $\mathfrak{so}(7)$ is special orthogonal}\label{section F4}

In this section, we show that the $8$-dimensional spinor representation $\mathbb{O}$ of $C^2({\rm Im}(\mathbb{O}),-q)$ is special orthogonal. Let $e_2,e_3,e_5 \in {\rm Im}(\mathbb{O})$ be such that $\mathcal{B}=\lbrace 1,e_2,e_3,e_2e_3,e_5,e_2e_5,e_3e_5,(e_2e_3)e_5 \rbrace$ is an orthogonal and anisotropic basis of $\mathbb{O}$ and set $e_4:=e_2e_3$, $e_6:=e_2e_5$, $e_7:=e_3e_5$, $e_8:=(e_2e_3)e_5$.
\vspace{0.2cm}

Since the Clifford algebra $C({\rm Im}(\mathbb{O}),-q)$ is $\mathbb{Z}_2$-graded, it is a Lie superalgebra for the bracket given by
$$\lbrace c,d \rbrace:=cd-(-1)^{|c||d|}dc \qquad \forall c,d \in C$$
where $|c|$ and $|d|$ denotes the parity of homogeneous elements $c$ and $d$. Let $\mathfrak{h}:=C^2({\rm Im}(\mathbb{O}),-q)$ and define the ad-invariant quadratic form $B_{\mathfrak{h}}$ on $\mathfrak{h}$ by
$$B_{\mathfrak{h}}(x,y)=-\frac{3}{8} Tr(\rho(x)\rho(y)) \qquad \forall x,y\in \mathfrak{h}.$$

\begin{df}
Let $\Omega:=Q(\eta^{-1}(\phi))\in C({\rm Im}(\mathbb{O}),-q)$. For $u\in {\rm Im}(\mathbb{O})$, define $c_u \in \mathfrak{h}$ by $c_u:=\lbrace u,\Omega \rbrace$ and define the $7$-dimensional subspace $W \subset \mathfrak{h}$ by $W:={\rm span}<\lbrace c_u ~ | ~ u \in {\rm Im}(\mathbb{O})\rbrace>$.
\end{df}

The subspace $W$ acts on $\mathbb{O}$ as follows.

\begin{pp} \label{action c_u}
Let $u,v \in {\rm Im}(\mathbb{O})$. We have
$$\rho(c_u)(1)=-6u, \quad \rho(c_u)(v)=2u\times v+6B(u,v).$$
\end{pp}

\begin{proof}
Using \eqref{decomp phi G2}, we obtain
$$\rho(\Omega)(1)=-7, \qquad \rho(\Omega)(u)=u,$$
and so
$$\rho(c_u)(1)=\rho(u)(\rho(\Omega)(1))+\rho(\Omega)(\rho(u)(1))=-7u+u=-6u.$$
Using \eqref{eq link product and cross product}, we have
\begin{align*}
    \rho(c_u)(v)&=\rho(u)(\rho(\Omega)(v))+\rho(\Omega)(\rho(u)(v))\\
    &=uv+\rho(\Omega)(uv)\\
    &=u\times v-B(u,v)+\rho(\Omega)(u\times v)-B(u,v)\Omega(1)\\
    &=2u\times v+6B(u,v).
\end{align*}
\end{proof}

Now, we can express the moment map of $\mathbb{O}$ in terms of the moment map of ${\rm Im}(\mathbb{O})$ and $W$:
\vspace{0.1cm}

\begin{pp}\label{pp mu so(7)}
The moment map $\mu_{\mathbb{O}}: \Lambda^2(\mathbb{O})\rightarrow \mathfrak{h}$ satisfies to
\begin{alignat}{2}
\mu_{\mathbb{O}}(u,v)&=\frac{8}{9}\mu_{{\rm Im}}(u,v)+\frac{1}{18}c_{u\times v} \qquad &&\forall u,v\in {\rm Im}(\mathbb{O}),\\
\mu_{\mathbb{O}}(u,1)&=\frac{1}{6}c_u \qquad &&\forall u\in {\rm Im}(\mathbb{O}).
\end{alignat}
\end{pp}

\begin{proof} We first need the following lemma:
\begin{lm}
Let $u,v\in {\rm Im}(\mathbb{O})$ and $D\in \mathfrak{g}$. We have
$$Tr(\rho(c_u)\rho(c_v))=-96B(u,v), \qquad Tr(\rho(D)\rho(c_u))=0.$$
\end{lm}
\begin{proof}
Let $w\in {\rm Im}(\mathbb{O})$. We have
\begin{align*}
        B(\rho(c_u)(\rho(c_v)(w)),w)&=2B(\rho(c_u)(v\times w),w)+6B(v,w)B(\rho(c_u)(1),w)\\
        &=2B(2u\times(v\times w)+6B(u,v\times w),w)-36B(u,w)B(v,w)\\
        &=-4B(u\times w,v\times w)-36B(u,w)B(v,w).
\end{align*}
The linearisation of \eqref{eq length cross product} gives
$$B(u\times w,v\times w)=B(u,v)q(w)-B(u,w)B(v,w)$$
and so
$$B(\rho(c_u)(\rho(c_v)(w)),w)=-4B(u,v)q(w)-32B(u,w)B(v,w).$$
We also have
$$B(\rho(c_u)(\rho(c_v)(1)),1)=-6B(\rho(c_u)(v),1)=-36B(u,v).$$
If $u$ and $v$ are orthogonal, without loss of generality (changing $\mathcal{B}$ if necessary), we can assume that $u=e_2$ and $v=e_3$. Hence
$$Tr(\rho(c_u)\rho(c_v))=\sum\limits_{e_i \in \mathcal{B}}\frac{1}{q(e_i)}B(\rho(c_u)(\rho(c_v)(e_i)),e_i)=0,$$
similarly we have
$$Tr(\rho(c_u)^2)=-96q(u)$$
and so
$$Tr(\rho(c_u)\rho(c_v))=-96B(u,v).$$
A straightforward calculation shows that $\mathfrak{g}$ and $W$ are orthogonal.
\end{proof}
Let $D\in \mathfrak{g}$.
We have
\begin{align*}
    B_{\mathfrak{h}}(D,\mu_{\mathbb{O}}(u,v))=B(D(u),v)=B_{\mathfrak{g}}(D,\mu_{{\rm Im}}(u,v))=\frac{8}{9}B_{\mathfrak{h}}(D,\mu_{{\rm Im}}(u,v)),
\end{align*}
and, using the previous lemma, we also have
$$B_{\mathfrak{h}}(c_w,\mu_{\mathbb{O}}(u,v))=B(c_w(u),v)=2B(w,u\times v)=-\frac{1}{48}Tr(c_wc_{u\times v})=\frac{1}{18}B_{\mathfrak{h}}(c_w,c_{u\times v}),$$
and so
$$\mu_{\mathbb{O}}(u,v)=\frac{8}{9}\mu_{{\rm Im}}(u,v)+\frac{1}{18}c_{u\times v}.$$
Since $\rho(D)(1)=0$, then we have $B_{\mathfrak{h}}(D,\mu_{\mathbb{O}}(u,1))=0$. Moreover,
$$B_{\mathfrak{h}}(c_w,\mu(u,1))=B(c_w(u),1)=6B(u,w)=-\frac{1}{16}Tr(c_wc_u)=\frac{1}{6}B_{\mathfrak{h}}(c_w,c_u)$$
and so
$$\mu_{\mathbb{O}}(u,1)=\frac{1}{6}c_u.$$
\end{proof}

The counterpart of $a)$ of Corollary \ref{expression mu g2 avec mucan} is the following property about the moment map of $\mathfrak{h}$:

\begin{cor} We have
\begin{equation}\label{eq moment map O of cyclic sum}
    \mu_{\mathbb{O}}(u,v\times w)+\mu_{\mathbb{O}}(v,w\times u)+\mu_{\mathbb{O}}(w,u\times v)=-\frac{1}{2}\mu_{\mathbb{O}}((u,v,w),1)\qquad \forall u,v,w\in {\rm Im}(\mathbb{O}).
\end{equation}
\end{cor}

\begin{proof}
Using $a)$ of Corollary \ref{expression mu g2 avec mucan}, we have
\begin{equation*}
    \mu_{\mathbb{O}}(u,v\times w)+\mu_{\mathbb{O}}(v,w\times u)+\mu_{\mathbb{O}}(w,u\times v)=\frac{1}{18}c_{u\times (v\times w)+v\times(w\times u)+w\times(u\times v)}.
\end{equation*}
Since, using \eqref{eq Jacobi G2}, we have
$$u\times (v\times w)+v\times(w\times u)+w\times(u\times v)=\frac{1}{4}J(u,v,w)=-\frac{3}{2}(u,v,w)$$
then we obtain
\begin{equation*}
    \mu_{\mathbb{O}}(u,v\times w)+\mu_{\mathbb{O}}(v,w\times u)+\mu_{\mathbb{O}}(w,u\times v)=-\frac{1}{12}c_{(u,v,w)}=-\frac{1}{2}\mu_{\mathbb{O}}((u,v,w),1).
\end{equation*}
\end{proof}

\begin{rem}
By Proposition \ref{pp mu so(7)}, the representation $\mathfrak{g}\rightarrow\mathfrak{so}(W,B_{\mathfrak{h}}|_W)$ of the quadratic Lie algebra $(\mathfrak{g},B_{\mathfrak{h}}|_{\mathfrak{g}})$ together with the non-trivial cubic term on $W$ given by a multiple of the cross-product is of Lie type in the sense of Kostant \cite{Kostant99}.
\end{rem}
\vspace{0.1cm}

We now give the main result of this section.
\vspace{0.1cm}

\begin{thm} \label{thm O is special}
The representation $\rho : \mathfrak{h} \rightarrow \mathfrak{so}(\mathbb{O},B)$ of the quadratic Lie algebra $(\mathfrak{h},B_{\mathfrak{h}})$ is a special orthogonal representation.
\end{thm}
\begin{proof}
We want to show Equation \eqref{eq CS}. Let $u,v,w\in {\rm Im}(\mathbb{O})$. We have
\begin{align*}
    \mu_{\mathbb{O}}(u,v)(w)+\mu_{\mathbb{O}}(u,w)(v)&=\frac{8}{9}(\mu_{\rm Im}(u,v)(w)+\mu_{\rm Im}(u,w)(v))+\frac{1}{18}(c_{u\times v}(w)+c_{u\times w}(v))\\
    &=-\frac{2}{9}([w,[u,v]]+[v,[u,w]])+\frac{1}{9}((u\times v)\times w+(u\times w)\times v)\\
    &=-\frac{8}{9}(w\times (u\times v)+v\times (u\times w))+\frac{1}{9}((u\times v)\times w+(u\times w)\times v)\\
    &=w\times(v\times u)+v\times (w\times u)
\end{align*}
and by Equation \eqref{eq Malcev id}, we obtain
$$\mu_{\mathbb{O}}(u,v)(w)+\mu_{\mathbb{O}}(u,w)(v)=B(u,v)w+B(u,w)v-2B(v,w)u$$
and so \eqref{eq CS} is satisfied for $u,v,w\in {\rm Im}(\mathbb{O})$. We have
\begin{align*}
    \mu_{\mathbb{O}}(u,v)(1)+\mu_{\mathbb{O}}(u,1)(v)&=\frac{1}{18}c_{u\times v}(1)+\frac{1}{6}c_u(v)=B(u,v),\\
    \mu_{\mathbb{O}}(1,v)(w)+\mu_{\mathbb{O}}(1,w)(v)&=-\frac{1}{6}(c_v(w)+c_v(w))=-2B(v,w),
\end{align*}
and so \eqref{eq CS} is satisfied whenever two elements $u,v$ or $w$ are in ${\rm Im}(\mathbb{O})$. Finally, since
$$ 2\mu_{\mathbb{O}}(u,1)(1)=-2u,\qquad \mu_{\mathbb{O}}(1,u)(1)=u,$$
then \eqref{eq CS} is satisfied whenever $u,v$ or $w$ is in ${\rm Im}(\mathbb{O})$ and so \eqref{eq CS} is satisfied for all $u,v,w\in \mathbb{O}$.
\end{proof}

By Theorems \ref{thm LSA from special orthogonal rep} and \ref{thm O is special} we have a Lie superalgebra $\tilde{\mathfrak{f}}$ of the form
$$\tilde{\mathfrak{f}}:=\mathfrak{h}\oplus \mathfrak{sl}(2,k)\oplus \mathbb{O} \otimes k^2.$$
This is an exceptional simple Lie superalgebra of type $F_4$ in the Kac notation.
\vspace{0.2cm}

\begin{rem}
If $k=\mathbb{R}$, Serganova (see \cite{Serganova1983}) showed that there are four real forms of $F_4$. In particular, two of them have an even part isomorphic to $\mathfrak{so}(7)\oplus \mathfrak{sl}(2,\mathbb{R})$ (resp. $\mathfrak{so}(4,3)\oplus \mathfrak{sl}(2,\mathbb{R})$) and an odd part isomorphic to the tensor product of the spinor representation of $\mathfrak{so}(7)$ (resp. $\mathfrak{so}(4,3)$) and $\mathbb{R}^2$. In our construction, if $\mathbb{O}$ is the compact or the split octonion algebra, both real forms of $F_4$ are obtained by our construction.
\end{rem}

Since the representation $\mathfrak{h} \rightarrow \mathfrak{so}(\mathbb{O},B)$ is special, we calculate its covariants $\psi_{\mathbb{O}}$, $Q_{\mathbb{O}}$ and the Mathews identities they satisfy. Since $\mathbb{O}$ is of dimension $8$, both-sides of Equations \eqref{id mat2} and \eqref{id mat3} vanish identically. However, both sides of the identities \eqref{id premat} and \eqref{id mat1} do no vanish identically. More precisely, up to constants, $\mu_{\mathbb{O}}\wedge_{\rho}\psi_{\mathbb{O}}$ and $Q_{\mathbb{O}}\wedge Id_{\mathbb{O}}\in {\rm Alt}_5(\mathbb{O},\mathbb{O})$ are the Hodge duals of the trilinear covariant $\psi_{\mathbb{O}} \in {\rm Alt}_3(\mathbb{O},\mathbb{O})$ and $\mu_{\mathbb{O}}\circ\psi_{\mathbb{O}}$ and $Q_{\mathbb{O}} \wedge \mu_{\mathbb{O}}\in {\rm Alt}_6(\mathbb{O},\mathfrak{h})$ are the Hodge duals of the moment map $\mu_{\mathbb{O}} \in {\rm Alt}_2(\mathbb{O},\mathfrak{h})$.
\vspace{0.2cm}

\begin{pp}
Let $\mu_{\mathbb{O}},\psi_{\mathbb{O}},Q_{\mathbb{O}}$ be the covariants of the special orthogonal representation $\rho : \mathfrak{h} \rightarrow \mathfrak{so}(\mathbb{O},B)$. We have
\begin{enumerate}[label=\alph*)]
\item $\psi_{\mathbb{O}}(v_1,v_2,v_3)=-\frac{1}{2}(v_1,v_2,v_3)+\phi(v_1,v_2,v_3)~$ and $~\psi_{\mathbb{O}}(v_1,v_2,1)=-v_1\times v_2~$  for all $v_1,v_2,v_3\in {\rm Im}(\mathbb{O})$.
\item $Q_{\mathbb{O}}(v_1,v_2,v_3,v_4)=\frac{2}{3}Q_{{\rm Im}}(v_1,v_2,v_3,v_4)~$ and $~Q_{\mathbb{O}}(v_1,v_2,v_3,1)=-4\phi(v_1,v_2,v_3)~$ for all $v_1,v_2,v_3,v_4\in {\rm Im}(\mathbb{O})$.
\item \begin{align}
        \eta^{-1}(Q_{\mathbb{O}})&=\frac{4}{q(e_2)q(e_3)}e_{1234}-\frac{4}{q(e_2)q(e_3)q(e_5)}e_{1278}+\frac{4}{q(e_2)q(e_3)q(e_5)}e_{1368}-\frac{4}{q(e_2)q(e_3)q(e_5)}e_{1467}\nonumber\\
        &+\frac{4}{q(e_2)q(e_5)}e_{1256}+\frac{4}{q(e_3)q(e_5)}e_{1357}+\frac{4}{q(e_2)q(e_3)q(e_5)}e_{1458}+\frac{4}{q(e_2)q(e_3)q(e_5)}e_{2358}\nonumber\\
        &-\frac{4}{q(e_2)q(e_3)q(e_5)}e_{2367}-\frac{4}{q(e_2)q(e_3)q(e_5)}e_{2457}-\frac{4}{q(e_2)^2q(e_3)q(e_5)}e_{2468}+\frac{4}{q(e_2)q(e_3)q(e_5)}e_{3456}\nonumber\\
        &-\frac{4}{q(e_2)q(e_3)^2q(e_5)}e_{3478}-\frac{4}{q(e_2)q(e_3)q(e_5)^2}e_{5678}.\label{decomp Q so(7)}
        \end{align}
\end{enumerate}
\end{pp}
\begin{proof}
$a)$ By Propositions \ref{pp mu so(7)} and \ref{pp covariants ImO}, we have
\begin{align*}
    \psi_{\mathbb{O}}(v_1,v_2,v_3)&=\frac{8}{9}\psi_{\rm Im}(v_1,v_2,v_3)+\frac{1}{18}(c_{v_1\times v_2}(v_3)+c_{v_2\times v_3}(v_1)+c_{v_3\times v_1}(v_1))\\
    &=-\frac{2}{3}(v_1,v_2,v_3)+\frac{1}{9}((v_1\times v_2)\times v_3+(v_2\times v_3)\times v_1+(v_3\times v_1)\times v_2)+\phi(v_1,v_2,v_3).
\end{align*}
Using Equations \eqref{eq link product and cross product} and \eqref{eq Jacobi G2}, we have
\begin{align*}
    (v_1\times v_2)\times v_3+(v_2\times v_3)\times v_1+(v_3\times v_1)\times v_2=-\frac{1}{4}J(v_1,v_2,v_3)=\frac{3}{2}(v_1,v_2,v_3)
\end{align*}
and so
$$\psi_{\mathbb{O}}(v_1,v_2,v_3)=-\frac{1}{2}(v_1,v_2,v_3)+\phi(v_1,v_2,v_3).$$
We also have
$$\psi_{\mathbb{O}}(v_1,v_2,1)=\frac{1}{18}c_{v_1\times v_2}(1)+\frac{1}{6}c_{v_2}(v_1)-\frac{1}{6}c_{v_1}(v_2)=-\frac{1}{3}v_1\times v_2+\frac{1}{3}v_2\times v_1-\frac{1}{3}v_1\times v_2=-v_1\times v_2.$$
\noindent
$b)$ Follows from $a)$ and Propositions \ref{pp formules lien BQ Bpsi dans le cas CS} and \ref{pp covariants ImO}.
\vspace{0.2cm}

\noindent
$c)$ Using $b)$, the decomposition of $Q_{\mathbb{O}}$ follows from the decompositions \eqref{decomp Q G2} and \eqref{decomp phi G2}.
\end{proof}

\begin{rem}
\begin{enumerate}[label=\alph*)]
\item We have
    $$Q_{\mathbb{O}}\wedge Q_{\mathbb{O}}(e_1\wedge\ldots\wedge e_8)=-224q(e_2)^2q(e_3)^2q(e_5)^2.$$
    If $char(k)\neq 7$, then $Q_{\mathbb{O}}\wedge Q_{\mathbb{O}}$ defines an orientation on $\mathbb{O}$
\item In the decomposition \eqref{decomp Q so(7)}, there are fourteen $4$-vectors of the form $e_{i_1}\wedge e_{i_2}\wedge e_{i_3}\wedge e_{i_4}$. The fourteen quadruples of indices $\lbrace i_1,i_2,i_3,i_4\rbrace$ appearing are not arbitrary. There is a numbering of the eight points of the affine space $(\mathbb{Z}_2)^3$ such that each quadruple corresponds to one of the fourteen affine planes.
\begin{center}
\begin{tikzpicture}
\draw [dashed] (0,0,0)--(2,0,0);
\draw [dashed] (0,0,0)--(0,2,0);
\draw (2,0,0)--(2,2,0);
\draw (2,2,0)--(0,2,0);
\draw (0,0,2)--(2,0,2)--(2,2,2)--(0,2,2)--cycle;
\draw [dashed] (0,0,0) -- (0,0,2);
\draw (2,0,0) -- (2,0,2);
\draw (2,2,0) -- (2,2,2);
\draw (0,2,0) -- (0,2,2);
\draw (0,0,0) node[below]{$3$};
\draw (0,0,2) node[below]{$1$};
\draw (0,2,0) node[above]{$7$};
\draw (0,2,2) node[above]{$5$};
\draw (2,0,0) node[below]{$4$};
\draw (2,0,2) node[below]{$2$};
\draw (2,2,0) node[above]{$8$};
\draw (2,2,2) node[above]{$6$};
\end{tikzpicture}
\end{center}
\end{enumerate}
\end{rem}

Suppose that $char(k)=0$ or $char(k)>7$. Define a quadratic form $B_{{\rm Alt}(\mathbb{O},\mathbb{O})}$ (resp. $B_{{\rm Alt}(\mathbb{O},\mathfrak{h})}$) on ${\rm Alt}_i(\mathbb{O},\mathbb{O})\cong\Lambda^i(\mathbb{O})^*\otimes \mathbb{O}$ (resp. ${\rm Alt}_i(\mathbb{O},\mathfrak{h})\cong\Lambda^i(\mathbb{O})^*\otimes \mathfrak{h}$) to be the tensor product of $B_{\Lambda^*}$ and $B$ (resp. $B_{\mathfrak{h}}$). For $f\in {\rm Alt}_i(\mathbb{O},\mathbb{O})$ define its Hodge dual $*f\in {\rm Alt}_{8-i}(\mathbb{O},\mathbb{O})$ to be the unique element which satisfies
$$\alpha\wedge_{B} *f=B_{{\rm Alt}(\mathbb{O},\mathbb{O})}(\alpha,f)Q_{\mathbb{O}}\wedge Q_{\mathbb{O}} \qquad \forall \alpha \in {\rm Alt}_i(\mathbb{O},\mathbb{O})$$
and for $f\in {\rm Alt}_i(\mathbb{O},\mathfrak{h})$ define its Hodge dual $*f\in {\rm Alt}_{8-i}(\mathbb{O},\mathfrak{h})$ to be the unique element which satisfies
$$\alpha\wedge_{B_{\mathfrak{h}}} *f=B_{{\rm Alt}(\mathbb{O},\mathfrak{h})}(\alpha,f)Q_{\mathbb{O}}\wedge Q_{\mathbb{O}} \qquad \forall \alpha \in {\rm Alt}_i(\mathbb{O},\mathfrak{h}).$$

\begin{pp}
We have
\begin{enumerate}[label=\alph*)]
\item $*\psi_{\mathbb{O}}=-56 Q_{\mathbb{O}}\wedge Id=\frac{112}{3} \mu_{\mathbb{O}}\wedge_{\rho}\psi_{\mathbb{O}},$
\item $*\mu_{\mathbb{O}}=-56 Q_{\mathbb{O}}\wedge \mu_{\mathbb{O}}=-\frac{56}{3} \mu_{\mathbb{O}}\circ\psi_{\mathbb{O}}.$
\end{enumerate}
\end{pp}

\begin{proof}
$a)$ Let $\alpha\in{\rm Alt}_3(\mathbb{O},\mathbb{O})$. We have
\begin{equation}\label{eq proof hodge so(7) 1}
    B_{{\rm Alt}(\mathbb{O},\mathbb{O})}(\alpha,\psi_{\mathbb{O}})Q_{\mathbb{O}}\wedge Q_{\mathbb{O}}(e_1\wedge \ldots \wedge e_8)=-224\sum\limits_{i<j<k}\frac{1}{q(e_i)q(e_j)q(e_k)}B(\alpha(e_i,e_j,e_k),\psi_{\mathbb{O}}(e_i,e_j,e_k))q(e_2)^2q(e_3)^2q(e_5)^2.
\end{equation}
On the other hand
\begin{align*}
    \alpha\wedge_B Q_{\mathbb{O}}\wedge Id(e_1\wedge \ldots \wedge e_8)&=\frac{1}{144}\sum\limits_{\sigma \in S_8}sgn(\sigma)B(\alpha(e_{\sigma(1)},e_{\sigma(2)},e_{\sigma(3)}),e_{\sigma(4)})Q_{\mathbb{O}}(e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)},e_{\sigma(8)}).
\end{align*}
Since $\alpha$ and $Q_{\mathbb{O}}$ are alternating and using the decomposition of Equation \eqref{decomp Q so(7)}, we have
\begin{equation}\label{eq proof hodge so(7) 2}
    \alpha\wedge_B Q_{\mathbb{O}}\wedge Id(e_1\wedge \ldots \wedge e_8)=\sum\limits_{\sigma \in S}sgn(\sigma)B(\alpha(e_{\sigma(1)},e_{\sigma(2)},e_{\sigma(3)}),e_{\sigma(4)})Q_{\mathbb{O}}(e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)},e_{\sigma(8)})
\end{equation}
where
$$S=\lbrace\sigma\in S_8 ~ | ~ \sigma(1)<\sigma(2)<\sigma(3),\quad (\sigma(5),\sigma(6),\sigma(7),\sigma(8))\in \lbrace (1,2,3,4),\ldots,(5,6,7,8) \rbrace \rbrace.$$
We have $|S|=56$ and each summand in \eqref{eq proof hodge so(7) 1} correspond to one summand in \eqref{eq proof hodge so(7) 2}. A straightforward calculation gives
\begin{align*}
    \alpha\wedge_B Q_{\mathbb{O}}\wedge Id(e_1\wedge \ldots \wedge e_8)&=4\sum\limits_{i<j<k}\frac{1}{q(e_i)q(e_j)q(e_k)}B(\alpha(e_i,e_j,e_k),\psi_{\mathbb{O}}(e_i,e_j,e_k))q(e_2)^2q(e_3)^2q(e_5)^2
\end{align*}
and so
$$\alpha\wedge_B Q_{\mathbb{O}}\wedge Id=-\frac{1}{56}B_{{\rm Alt}(\mathbb{O},\mathbb{O})}(\alpha,\psi_{\mathbb{O}})Q_{\mathbb{O}}\wedge Q_{\mathbb{O}}.$$

\noindent
$b)$ Let $\alpha\in{\rm Alt}_2(\mathbb{O},\mathfrak{h})$. We have
$$B_{{\rm Alt}(\mathbb{O},\mathfrak{h})}(\alpha,\mu_{\mathbb{O}})Q_{\mathbb{O}}\wedge Q_{\mathbb{O}}(e_1\wedge\ldots\wedge e_8)=-224\sum\limits_{i<j}\frac{1}{q(e_i)q(e_j)}B_{\mathfrak{h}}(\alpha(e_i,e_j),\mu_{\mathbb{O}}(e_i,e_j))q(e_2)^2q(e_3)^2q(e_5)^2.$$
On the other hand
\begin{align*}
    \alpha\wedge_{B_{\mathfrak{h}}} Q_{\mathbb{O}}\wedge \mu_{\mathbb{O}}(e_1\wedge \ldots\wedge e_8)&=\frac{1}{96}\sum\limits_{\sigma\in S_8}sgn(\sigma)B_{\mathfrak{h}}(\alpha(e_{\sigma(1)},e_{\sigma(2)}),\mu_{\mathbb{O}}(e_{\sigma(3)},e_{\sigma(4)}))Q_{\mathbb{O}}(e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)},e_{\sigma(8)}).
\end{align*}
Since $\alpha, \mu_{\mathbb{O}}$ and $Q_{\mathbb{O}}$ are alternating and using the decomposition of Equation \eqref{decomp Q so(7)}, we have
$$\alpha\wedge_{B_{\mathfrak{h}}} Q_{\mathbb{O}}\wedge \mu_{\mathbb{O}}(e_1\wedge \ldots\wedge e_8)=\sum\limits_{\sigma\in S}sgn(\sigma)B_{\mathfrak{h}}(\alpha(e_{\sigma(1)},e_{\sigma(2)}),\mu_{\mathbb{O}}(e_{\sigma(3)},e_{\sigma(4)}))Q_{\mathbb{O}}(e_{\sigma(5)},e_{\sigma(6)},e_{\sigma(7)},e_{\sigma(8)})$$
where
$$S=\lbrace\sigma\in S_8 ~ | ~ \sigma(1)<\sigma(2),\quad \sigma(3)<\sigma(4),\quad (\sigma(5),\sigma(6),\sigma(7),\sigma(8))\in \lbrace (1,2,3,4),\ldots,(5,6,7,8) \rbrace \rbrace.$$
We have $|S|=84$ and, using Equation \eqref{eq moment map O of cyclic sum}, a straightforward calculation gives
$$\alpha\wedge_{B_{\mathfrak{h}}} Q_{\mathbb{O}}\wedge \mu_{\mathbb{O}}(e_1\wedge \ldots\wedge e_8)=4\sum\limits_{i<j}\frac{1}{q(e_i)q(e_j)}B_{\mathfrak{h}}(\alpha(e_i,e_j),\mu_{\mathbb{O}}(e_i,e_j))q(e_2)^2q(e_3)^2q(e_5)^2,$$
and so
$$\alpha\wedge_{B_{\mathfrak{h}}} Q_{\mathbb{O}}\wedge \mu_{\mathbb{O}}(e_1\wedge \ldots\wedge e_8)=-\frac{1}{56}B_{{\rm Alt}(\mathbb{O},\mathfrak{h})}(\alpha,\mu_{\mathbb{O}})Q_{\mathbb{O}}\wedge Q_{\mathbb{O}}.$$
\end{proof}

\begin{rem}
One can show similarly that, up to a constant,
the identity is the Hodge dual of $Q_{\mathbb{O}}\wedge \psi_{\mathbb{O}}$.
\end{rem}

\footnotesize
\bibliographystyle{alpha}


\section{Matrix Concentration via Doob Martingale}
Our concentration proof proceeds by constructing a Doob martingale and controlling the norm of each increment and the total predictable variation of the martingale process. Let $$Y_k = \E[f(X_1, ..., X_n) | X_1, ..., X_k] - \E[f(X_1, ..., X_n) | X_1, ..., X_{k-1}],$$ where $f(X_1, ..., X_n) =\prod\limits_{i=1}^{n}\big(I +\frac{X_i}{n}\big)$. Note that $\E[Y_i |  X_1, ..., X_i] = 0$, thus $Y_i$ is a martingale. We also observe that as $X_1, ..., X_n$ are independent, \begin{align*}Y_k &= \E\left[f(X_1, ..., X_n) | X_1, ..., X_k\right] - \E\left[f(X_1, ..., X_n) | X_1, ..., X_{k-1}\right] \\
&=  \prod_{i=1}^k \big(I+\frac{X_i}{n}\big) \prod\limits_{i=k+1}^{n}\E\bigg[\big(I +\frac{X_i}{n}\big)\big] - \prod_{i=1}^{k-1} \big(I + \frac{X_i}{n}\big) \prod\limits_{i=k+1}^{n}\E\bigg[\big(I +\frac{X_i}{n}\big)\bigg] \\
&= \prod_{i=1}^{k-1} \big(I+\frac{X_i}{n}\big) \frac{X_k - \mu
}{n}\prod\limits_{i=k+1}^{n}\big(I +\frac{\mu}{n}\big).
\end{align*}
We thus use submultiplicativity of the spectral norm to obtain,
\begin{align*}
    \|Y_k\|&=\bigg\|\prod_{i=1}^{k-1} \big(I+\frac{X_i}{n}\big) \cdot \frac{X_k - \mu
}{n}\cdot\prod\limits_{i=k+1}^{n}\big(I +\frac{\mu}{n}\big)\bigg\|\\
&\leq \bigg(\prod\limits_{i=1}^{k-1}\bigg\|I+\frac{X_i}{n}\bigg\|\bigg)\bigg\|\frac{X_k-\mu}{n}\bigg\|\bigg(\prod\limits_{i=k+1}^{n}\bigg\|\bigg(I+\frac{\mu}{n}\bigg)\bigg\|\bigg)\\
&\leq \frac{2L}{n}\big(1+\frac{L}{n}\big)^{n-1}\\
&\leq \frac{2Le^L}{n},
\end{align*}
where the second inequality follows from the norms of $X_i$ (and hence norm of $\mu$) being bounded by $L$ almost surely and the last inequality follows as $(1+x/n)^{(n-1)}\leq (1+x/n)^n\leq e^x$ for non-negative $x$.

Also note that \begin{align*}\left\|\mathbb{E}\left[Y_k Y_k^*| X_1,\ldots,X_{k-1}\right]\right\| &= \left\|\left(\prod_{i=1}^{k-1} I+\frac{X_i}{n}\right) \frac{X_k - \mu
}{n}\prod\limits_{i=k+1}^{n}\left(I +\frac{\mu}{n}\right)\prod\limits_{i=n}^{k+1}\left(I +\frac{\mu}{n}\right)\frac{X_k^* - \mu
}{n}\left(\prod_{i=k-1}^{1} I+\frac{X_i^*}{n}\right)\right\| \\
&≤ \prod_{i=1}^{k-1} \left\|I+\frac{X_i}{n}\right\| \cdot \left\|\frac{X_k - \mu
}{n}\right\| \prod\limits_{i=k+1}^{n}\left\|I +\frac{\mu}{n}\right\| \prod\limits_{i=n}^{k+1}\left\|I +\frac{\mu}{n}\right\| \cdot \left\|\frac{X_k^* - \mu
}{n}\right\|\prod_{i=k-1}^{1} \left\|I+\frac{X_i^*}{n} \right\|  \\
&≤ \frac{4L^2}{n^2} \left(1+\frac{L}{n}\right)^{2n-2} \\
&≤ \frac{4L^2}{n^2} e^{2L}. \\
\end {align*}
Hence, we get that for any $k\leq n$,
\begin{align*}
    \left\|\sum\limits_{i=1}^{k}\mathbb{E}\left[Y_k Y_k^*| X_1,\ldots,X_{k-1}\right]\right\|
    &\leq \sum\limits_{i=1}^{k}\left\|\mathbb{E}\left[Y_k Y_k^*| X_1,\ldots,X_{k-1}\right]\right\|\\
    &\leq \frac{4L^2e^{2L}k}{n^2}\\
    &\leq\frac{4L^2e^{2L}}{n}.
\end{align*}
To conclude the proof, we use the Matrix Freedman inequality \cite{TroppIntro15} for concentration of matrix valued martingales which is stated next.
\begin{theorem}\label{thm:freedman}
Suppose $Y_k=\sum\limits_{i=1}^k X_i $ is a martingale with $d\times d$ matrix increments $X_i$ satisfying $\|X_i\|\leq R$ almost surely. Let the predictable variations of the process be $W_k^{(1)}=\sum\limits_{i=1}^{k}\E[X_i X_i^*|X_1,\ldots,X_{i-1}]$ and $W_k^{(2)}=\sum\limits_{i=1}^{k}\E[X_i^* X_i|X_1,\ldots,X_{i-1}]$. Then for all $t\geq 0$, we have
\begin{align*}
    \mathsf{Pr}[\exists k \geq 0 : \|Y_k\|\geq t \ \mathit{and} \   \max\{\|W_k^{(1)}\|,\|W_k^{(2)}\|\}\leq \sigma^2]\leq 2d\exp\left(-\frac{c t^2}{Rt + \sigma^2}\right).
\end{align*}
\end{theorem}

\begin{proof}[Proof of Theorem \ref{thm:mainthm}]
From the above argument, we get that the increments of our martingale $Y_k$ are bounded by $Le^L/n$ in spectral norm almost surely and that the norm of the predictable quadratic variation (the analysis of $\mathbb{E}[Y_k^* Y_k| X_1,\ldots,X_{k-1}]$ is identical) is bounded by $\frac{4L^2e^{2L}}{n}$ almost surely. Hence we can use Thereom \ref{thm:freedman}, to conclude that

\begin{align*}
     \mathsf{Pr}\left[\|Y_n\|\geq t \right]&\leq 2d\exp\left(-\frac{cnt^2}{Le^Lt + L^2e^{2L}}\right)\\
     &\leq 2d\exp\left(-\frac{cnt^2}{2L^2e^{2L}}\right),
\end{align*}
where for the second inequality we have assumed that $t\leq Le^L\sqrt{\frac{\log d}{n}}\leq Le^L.$

\end{proof}

\section{Setup}
Suppose ${X_1},\ldots,{X_n} \in \mathbb C^{d\times d}$ are random matrices sampled i.i.d from some distribution with $\E[{X_i}]=\mu$ and $\|{X}_i\|_\mathsf{op}\leq L$ almost surely.
A famous result is the matrix Bernstein inequality \cite{TroppIntro15} for sums of random matrices, which in this setting asserts that
\begin{align*}
    \mathsf{Pr}\left[\left\|\sum\limits_{i=1}^{n}\frac{X_i}{n}-\mu\right\|_\mathsf{op}\geq t\right] \leq 2d \cdot \exp(-n t^2/2L^2),
\end{align*}
whenever $t \leq L\sqrt{\frac{\log d}{n}}$ and $n\geq \log(d)$. For some numerical linear algebra problems, it is of interest to consider instead of sums, functions of the form $$f({X_1},\ldots, {X_n}) = \prod\limits_{i=1}^{n}\left({I} +\frac{{X}_i}{n}\right).$$ We will refer to such functions as matrix product functions. One can easily prove the following lemma
\begin{lemma}$\E_{{X_1},\ldots,{X_n}}[f(X_1,\ldots,X_n)] \preceq e^\mu$ with equality in the limit as $n\rightarrow \infty$.
\end{lemma}
\begin{proof}
\begin{align*}
    \E_{{X_1},\ldots,{X_n}}[f({X_1},\ldots,{X_n})] &=  \E_{{X_1},\ldots,{X_n}}\left[\prod\limits_{i=1}^{n}\left(\mathbf{I} +\frac{{X}_i}{n}\right)\right]\\
    &=\prod\limits_{i=1}^{n}\E_{{X_i}}\left[{I}+\frac{{X_i}}{n}\right]\\
    &=\prod\limits_{i=1}^{n}\left[{I}+\frac{\mu}{n}\right]\\
    &=\left({I}+\frac{\mu}{n}\right)^n
    \preceq e^\mu,
\end{align*}
and there is equality in the limit. The second equality is because of independence of ${X_i}.$
\end{proof}
Recently a central limit theorem for matrix products was established \cite{EH18} and the following concentration inequality was proven by Henriksen and Ward \cite{HW19}.
\begin{theorem}[\cite{HW19}]Assuming $\max\{3,Le^2\}\leq \log(n)+1\leq \big(\frac{16n}{\log(dne/\delta)}\big)^{1/3}$, we have that with probability greater than $1-2\delta$, the following holds
\begin{align*}
    \|f({X_1,\ldots,X_n)}-e^\mu\|\leq \frac{O(Le^L)\log(n)}{\sqrt{n}}\big(\sqrt{\log(d/\delta)+\log(n)^2}+\frac{\log(n)}{\sqrt{n}}\big)+\frac{L^2e^L}{n}.
\end{align*}
\end{theorem}

 Their proof groups the product into sums of $k-$wise products in a careful way, appealing to Baranyai's theorem, and applies matrix Bernstein inequality to each partition. This approach loses a $(\log n)^2$ factor compared to the matrix Bernstein result for sums and it is unclear whether this is necessary. In this note, we will give a simple proof relying on the Matrix Freedman inequality \cite{TroppIntro15} which does not lose the $\log n$ factors, essentially matching the matrix Bernstein inequality for sums of matrices upto constants.
\begin{theorem}\label{thm:mainthm}\begin{align*}\mathsf{Pr}\left[\left\|f({X}_1,\ldots,{X}_n)-e^\mu\right\|_\mathsf{op}\geq t\right] \leq 2d \cdot \exp(-cn t^2/L^2e^{2L}),\end{align*}
whenever $t\leq Le^L\sqrt{\frac{\log d}{n}}$, for some absolute constant $c$. Equivalently, for every $\delta\in(0,1)$ with probabiity greater than $1-\delta$, we have
\begin{align*}
    \|f({X_1,\ldots,X_n)}-e^\mu\|\leq \frac{O(Le^L)}{\sqrt{n}}\sqrt{\log(d/\delta)}.
\end{align*}
\end{theorem}
The key difference in this result and the matrix Bernstein inequality for sums is the $L^2e^{2L}$ factor instead of $L^2$. We will later show that even for the special case of products of scalars, such an $e^{O(L)}$ dependence is necessary if the bound is written only in terms of $L$ and not $\mu$.
\begin{remark}[Independent Work]
The recently posted independent work \cite{newprod} gives a different proof of a more refined version of Theorem \ref{thm:mainthm}, which has slightly better constants and an $L^2e^{2\mu}$ term in the denominator rather than $L^2e^{2L}$ (see their Theorem I). Their approach is also martingale-based, but instead of Matrix Freedman it relies on certain smoothness properties of Schatten norms, also yielding more general results for Schatten norms of matrix products which our proof does not yield.
\end{remark}
\input{martingale.tex}

\section{Lower Bound}
In this section, we show that the tail bound needs to depend as $L^2e^{O(L)}$ as given in Theorem \ref{thm:mainthm} even for the case of scalars rather than matrices. Consider a two-point distribution which takes values $X_i = 0$ or $X_i = 2L$ with equal probability. $X_i$ can thus be represented as $X_i = L+LY_i$ where $Y_i$ is a Rademacher random variable. Thus $\mathbb{E}[X] = L$. For sufficiently large $n$, $\prod\limits_{i=1}^{n}\left(1+\frac{X_i}{n}\right)= exp\left(\sum\limits_{i=1}^{n}\frac{X_i}{n}\right)(1+o_n(1))$. Taking $t=Le^Lc$, we have:
\begin{align*}
    \mathsf{Pr}\left[exp\left(\sum\limits_{i=1}^{n}\frac{L+LY_i}{n}\right)-e^L\geq cLe^L\right]&=\mathsf{Pr}\left[exp\left(\sum\limits_{i=1}^{n}\frac{LY_i}{n}\right)-1\geq cL\right]\\
    &=\mathsf{Pr}\left[\sum\limits_{i=1}^{n}\frac{LY_i}{n}\geq \log(1+cL)\right]\\
    &\geq \mathsf{Pr}\left[\sum\limits_{i=1}^{n}\frac{LY_i}{n}\geq cL\right]\\
    &\geq \mathsf{Pr}\left[\sum\limits_{i=1}^{n}\frac{Y_i}{n}\geq c\right],
\end{align*}
where the first inequality follows as $\log(1+x) < x$ for sufficiently large $x$ and hence corresponds to a larger probability event. Hence, we obtain a lower bound on the probability which is independent of $L$ and so indeed the $Le^{O(L)}$ term must appear in the tail bound. Here we have $O(L)$ in the exponent because in the lower bound example, the $X_i$ are bounded by $2L$ rather than $L$.


\section{Introduction}

Planetary nebulae (PNe) are the descendants of low- and intermediate-mass
stars, as they eject their envelopes at the tip of the asymptotic giant
branch (AGB) and start their evolution towards the white dwarf phase.
At this time, the central star of the PN (CSPN) develops a fast (although
tenuous) stellar wind that impinges and snowplows the material previously
ejected during the AGB as a slow and dense stellar wind.
In this interacting stellar winds (ISW) model for the formation
and evolution of PNe, the fast stellar wind is shock-heated up
to temperatures that depend on the CSPN terminal wind velocity
as $T\sim v_{\infty}^2$ \citep{Dyson1997}.
For typical CSPN wind velocities of $v_{\infty} = 500-4000$~km~s$^{-1}$
\citep[e.g.,][]{Guerrero2013}, shocked plasma at X-ray-emitting temperatures
in excess of 10$^6$~K can be expected.

The very first \emph{Chandra} observations of PNe already found hot
bubbles with temperatures of a few million K confined within the
inner rims of BD$+$30$^{\circ}$3639 \citep{Kastner2000} and NGC\,6543
\cite{Chu2001}.
This general behavior has been largely confirmed by the \emph{Chandra}
Planetary Nebula Survey \citep[ChanPlaNS;][]{Kastner2012,Freeman2014},
a volume-limited survey of a sample of nearby ($d<1.5$~kpc) PNe.
Diffuse X-ray emission is detected only in compact ($r<0.2$~pc)
relatively young PNe with a closed-shell morphology \citep[as
suggested previously by][]{Ruiz2011}, for a detection rate
$\sim$30\%.

\emph{Chandra} has also unveiled unexpected sources of hard X-ray emission
($>$0.5 keV) from CSPNe \citep{Guerrero2001}, and ChanPlaNS has confirmed
the prevalence of these hard X-ray point-sources in a significant
number of CSPNe \citep{Montez2015}.
Some of these hard X-ray sources can be attributed to the coronal
emission from a dwarf or late-type giant companion \citep{Montez2010},
while others can be assigned to shocks in their fast winds as in OB
and Wolf-Rayet stars, especially among CSPNe with powerful fast stellar
winds \citep{Guerrero2001}.
The X-ray to bolometric luminosity ratios of the latter,
$L_{\rm X}/L_{\rm  bol} \sim 10^{-7}$, are indeed similar to
those of single OB stars \citep[e.g.,][]{Nebot2018}.

\section{Available X-ray Observatories}

\emph{ROSAT} obtained pointed or serendipituous observations of 63 PNe,
but only 16 were detected and most of these detections correspond to
soft photospheric emission from hot CSPNe, whereas only three, namely
BD$+$30$^{\circ}$3639, NGC\,6543, and NGC\,7009 could be tentatively
attributed to diffuse hot gas \citep{Guerrero2000}.
Most modern X-ray observations of PNe have been obtained using the
\emph{Chandra} and \emph{XMM-Newton} observatories, which were launched
in 1999, i.e., more than 20 years ago!
Still, they have been used rather than other more modern X-ray observatories
such as \emph{Swift} or \emph{AstroSat} because of the much larger effective
areas of \emph{Chandra} and \emph{XMM-Newton} in the energy range of interest
($\leq$1.5 keV) for PNe.
Specifically, the effective area of the \emph{Chandra} ACIS-S7 and
\emph{XMM-Newton} EPIC-pn detectors at 1.5 keV are 5--8 and 9--13 times
larger, respectively, than those of \emph{Swift} XRT ($\simeq$135 cm$^2$)
and \emph{AstroSat} ($\simeq$90 cm$^2$).

\begin{figure}[t]
\centering
\includegraphics[bb=18 220 600 640,width=0.75\columnwidth]{stat.eps}
\caption{
(bottom)
Time evolution of the number of PNe detected by the \emph{Chandra} (red)
and \emph{XMM-Newton} (blue) X-ray observatories.
(top)
Time evolution of the number of papers (black) and citations to these papers
(blue) based on those X-ray observations (source Astronomycal Database System,
ADS, at https://ui.adsabs.harvard.edu).
Note that the left and right axes in this panel refer to numbers of
papers and citations, respectively, as denoted by their labels and
colors.
The red-dashed line is a linear fit to the number of papers, implying
a publication rate $\simeq$3 papers per year on the X-ray emission
from PNe.
}
\label{fig.stat}
\end{figure}

Since \emph{Chandra} and \emph{XMM-Newton} started operations in 1999,
the number of PNe with diffuse X-ray emission and CSPNe with hard X-ray
emission from grown steadily.
As illustrated in the bottom panel of Figure~\ref{fig.stat},
\emph{Chandra} and \emph{XMM-Newton} have revealed these types
of X-ray emission in almost 40 PNe.
The notable increase of PNe between 2010 and 2015 can be attributed to
the effort of the PN community to obtain the ChanPlaNS survey, which has
resulted in unprecedented progress in the understanding of the nature
of point and extended sources of X-rays in PNe.
The impact of all these observations has been tremendous.
Almost 60 papers have been written in this topic since the launch of
\emph{Chandra} and \emph{XMM-Newton} at a publication rate of three
papers per year (top panel of Figure~\ref{fig.stat}).
By the end of 2019, these papers had garnered close to 1,500 citations.

The bottom panel of Figure~\ref{fig.stat} reveals a worrisome fact.
Despite its unbeatable spatial resolution, \emph{Chandra} has not
contributed to any new X-ray detection among PNe since 2014
(\emph{XMM-Newton} has made two new discoveries since then).
This can be mostly attributed to
the continuous degradation of \emph{Chandra} ACIS's sensitivity to low
energy photons, which has made it more and more unlikely to detect the
soft X-ray emission from PNe with sufficient photon counts for detailed
spectral analysis\footnote{
  The sensitivity of \emph{Chandra} HRC has remained basically unchanged
  in this time period, but the spectral capability of this instrument,
  which is otherwise very well suited for high-resolution imaging studies,
  is very limited.}.
This is illustrated in Figure~\ref{fig.acis}, showing the time evolution
of the exposure time required for a \emph{Chandra} ACIS-S observation to
obtain the same number of counts for a plasma model typical of the diffuse
X-ray emission in PNe.
Similarly, the simulation of the \emph{Chandra} ACIS-S count number and
image of BD+30$^\circ$3639 through Cycle 10 ($\simeq$2500 cnts), Cycle 15
($\simeq$1500 cnts) and Cycle 20 ($\simeq$350 cnts) compares very poorly
to those observed in Cycle 1 ($\simeq$4400 cnts, Montez, priv.\
communication).
In this sense, the ChanPlaNS survey was very timely,
using the last useful \emph{Chandra} window to acquire
imaging spectroscopic observations of PNe.

\begin{figure}[t]
\centering
\includegraphics[bb=25 230 585 535,width=0.75\columnwidth]{acis.eps}
\caption{
Time evolution of the exposure time required for a \emph{Chandra} ACIS-S
observation to obtain a given number of counts for a plasma model typical
of the diffuse X-ray emission in PNe.
Simulations were obtained using the \emph{Chandra} PIMMS v4.9
simulator, which includes the most up-to-date calibrations.
A plasma with a typical 1.6$\times$10$^6$ K temperature,
low hydrogen column density, and slightly subsolar chemical
composition has been assumed.
}
\label{fig.acis}
\end{figure}

\section{Latest Results on X-ray Emission from PNe}

Since the WORPLANS~I meeting in December 2015, a number of new results
based on X-ray observations or theoretical investigations have been
reported.
These can be grouped in four different categories.

\subsection{Theoretical Studies}

The large amount of observational data already available has allowed the
refinement of our ISW models on the production of X-ray-emitting gas in
PNe.
Two different research lines have been pursued.
The special properties of the hot bubbles produced by CSPNe with H-deficient
stellar winds has been extensively investigated, with particular attention
paid to the effects of heat conduction \citep{Sandin2016} and its application
to the case of BD+30$^\circ$3639 \citep{Heller2018}.
On the other hand, the combined effects of heat conduction and turbulent
mixing have been tested using high-resolution, two-dimensional,
radiation-hydrodynamical simulations \citep{Toala2016,Toala2018}.
In these detailed models, instabilities in the wind-wind interaction zone
produce clumps and filaments in the swept-up shell of nebular material
with notable effects in the time-dependent X-ray emission as the stellar
parameters change.
It is shown that the diffuse X-ray emission at early times is dominated
by the contribution from the hot, shocked stellar wind, whereas the
contribution from nebular gas dominates at later times.
Furthermore, these models demonstrate that the X-ray temperature of a hot
bubble is anti-correlated with metallicity, but its X-ray luminosity increases
with metallicity.
The conclusions are quite robust, turbulent mixing layers are the origin
of the soft X-ray emission in the majority of diffuse nebulae.

\subsection{Multi-wavelength Studies}

Multi-wavelength studies combining X-ray and infrared, optical, and UV
observations have provided very interesting results in the past few years.
The obvious correlation between the X-ray morphology and the varying local
extinction, as the soft X-ray emission from PNe is easily absorbed by even
small amounts of material along the line of sight, has been confirmed through
the comparison of optical and infrared images of BD+30$^\circ$3639
\citep{FK2016} and NGC\,7027 \citep{MK2018} with archival or new
\emph{Chandra} observations.
The discovery of the mixing layer at the interface between the X-ray-emitting
hot bubble and the optical photo-ionized nebular shell in NGC\,6543 using
\emph{HST} STIS observations \citep{Fang2016} is very exciting.
The emission in the N~{\sc v} $\lambda\lambda$1239,1243 \AA\ lines
from this interface layer has allowed for the first time not only
to probe it, but to determine its spatial extent, thickness and
physical properties.
These are critical to assess the relative importance of heat conduction and
turbulent mixing in the production of hot gas in PNe.

\subsection{X-ray Variability Studies}

The hard X-ray emission from the CSPN of the Eskimo Nebula has been found
to be variable with a period $\simeq$6 hours \citep{Guerrero2019}.
This X-ray emission might be attributed to accretion of material from the
CSPN wind onto a white dwarf companion, and thus the observed period can
be assigned either to the orbital period of the companion or to the rotation
of an accretion disk around it.
Time-analysis of the X-ray emission from CSPNe can be
very revealing of the nature of putative companions.

\subsection{New Discoveries of X-ray PNe}

Only one more PN has been found to exhibit diffuse X-ray emission since
the WORKPLANS~I.
This is the case of NGC\,5189, a PN around a [WO1] CSPN whose extended
X-ray emission has been discovered by \emph{XMM-Newton} \citep{Toala2019}.
The spatial distribution and spectral properties of the X-ray emission,
particularly its high carbon abundances, are consistent with a born-again
scenario where a CSPN becomes carbon-rich through a very late thermal pulse
(VLTP) event and then develops a fast, carbon-rich wind that generates X-ray
emission.
The physics of the interaction between the present fast stellar
wind of the CSPN and the hydrogen-poor clumps of material ejected in the
VLTP event is fascinating \citep[see also the cases of A\,30 and
A\,78,][]{Guerrero2012,Toala2015}.

\section{The Future of X-ray Observations of PNe}

In the long run,
the \emph{X-ray Imaging and Spectroscopic Mission} (\emph{XRISM}),
the \emph{X-ray Universe Baryon Surveyor} (\emph{HUBS}),
the \emph{Advanced Telescope for High ENergy Astrophysics} (\emph{Athena}),
and
the \emph{Lynx X-ray Observatory} (\emph{Lynx}),
with their enhanced collecting areas, high-dispersion instruments, and/or
spatial resolution will play major roles in a better understanding of the
production of hot gas in the interior of PNe and its role in their evolution
and shaping.
Meanwhile, the \emph{eROSITA} mission will produce an X-ray map of the
whole sky with unprecedented sensitivity.
The final \emph{eROSITA} All Sky Survey (eRASS) is expected to
have a sensitivity in the range from 1$\times$10$^{-14}$ down
to 3$\times$10$^{-17}$ erg~cm$^{-2}$~s$^{-1}$, depending on the
final exposure time of a particular position on the sky.

We have convolved the expected sensitivity map of the final eRASS with the
position of all known PNe to determine their exposure times and X-ray
sensitivity.
The expected distributions of exposure times and sensitivities for
the whole sample of known PNe are shown in Figure~\ref{fig.erass}.
It reveals that $\simeq$200 PNe will be observed with exposure
times greater than 20 ks.
As the seven X-ray telescopes of \emph{eROSITA} have a total effective area
above that of \emph{XMM-Newton} EPIC-pn, almost 500 PNe will have observations
as sensitive, $\leq$2$\times$10$^{-15}$ erg~cm$^{-2}$~s$^{-1}$, as those of PNe
in the ChanPlaNS survey.
Obviously this will produce a major improvement in our statistics of the
presence of X-ray emission in PNe.

\begin{figure}[t]
\centering
\includegraphics[bb=25 310 585 500,width=0.9\columnwidth]{eRASS.eps}
\caption{
Distribution of exposure time (left) and sensitivity (rigth) for PNe
expected in the final eRASS map of the whole sky.
Most PNe will be located in areas of the eRASS with total exposure times
shorter than 20 ks, but a significant number of them, $\simeq$200, will
have longer exposure times.
Accordingly, a significant number of PNe, those with long
exposure times in the final eRASS, will have observations
with sensitivities adequate to detect X-ray fluxes smaller
than $\simeq$2$\times$10$^{-15}$ erg~cm$^{-2}$~s$^{-1}$.
}
\label{fig.erass}
\end{figure}

\section{Conclusions}

Since the advent of the modern X-ray observatories \emph{Chandra} and
\emph{XMM-Newton}, the number of PNe with diffuse X-ray emission and
CSPNe with hard X-ray emission have grown steadily.
The large number and quality of these X-ray data sets of PNe
has resulted in major advances in a research field for which
sensitive observations basically did not exist before 1999.
Over 60 papers have been generated
based on these observations since 1999,
garnering almost 1,500 citations.

This publication rate has remained since the last WORKPLANS meeting in
December 2015, but the emphasis of the studies has changed significantly.
In this time period, notable advances in theoretical analyses have
been made and multi-wavelength and time-domain studies are gaining
momentum, while the discoveries of new cases for X-ray emission from PNe
or their CSPNe has slowed nearly to a halt, mostly due to the dramatic
decrease of sensitivity to soft X-ray photons of \emph{Chandra} ACIS.

In the near future, \emph{eROSITA} can make a major contribution
to our understanding of the production of hot gas in PNe and its
effects in their evolution and shaping.
Hundreds of PNe can be expected to be observed with sensitivities
similar to many of the currently available observations carried
out by \emph{Chandra} and \emph{XMM-Newton} in the last two decades.

\vspace{6pt}

\funding{
This research was funded by the Spanish Ministerio de Ciencia, Innovaci\'on
y Universidades grant number PGC2018-102184-B-I00 co-funded by FEDER funds.
}

\acknowledgments{
The author acknowledges the scientific discussions with
Joel H.\ Kastner, Rodolfo Montez, Quentin Parker, and
Jes\'us A.\ Toal\'a, which helped him assess the impact of
\emph{eROSITA} and its eRASS for the research of X-ray
emission from PNe.
Four anonymous referees are also acknowledged for their useful comments
and suggestions to improve this manuscript.
}

\conflictsofinterest{The authors declare no conflict of interest.}

\reftitle{References}