Title: A characterisation of Lie algebras amongst anti-commutative algebras,
Abstract: Let $\mathbb{K}$ be an infinite field. We prove that if a variety of
anti-commutative $\mathbb{K}$-algebras - not necessarily associative, where
$xx=0$ is an identity - is locally algebraically cartesian closed, then it must
be a variety of Lie algebras over $\mathbb{K}$. In particular,
$\mathsf{Lie}_{\mathbb{K}}$ is the largest such. Thus, for a given variety of
anti-commutative $\mathbb{K}$-algebras, the Jacobi identity becomes equivalent
to a categorical condition: it is an identity in~$\mathcal{V}$ if and only if
$\mathcal{V}$ is a subvariety of a locally algebraically cartesian closed
variety of anti-commutative $\mathbb{K}$-algebras. This is based on a result
saying that an algebraically coherent variety of anti-commutative
$\mathbb{K}$-algebras is either a variety of Lie algebras or a variety of
anti-associative algebras over $\mathbb{K}$.