On [this answer](https://math.stackexchange.com/a/2192799/396515) about divergence of a cross product, the following proof using Einstein notation appears:

\begin{eqnarray\*}
 \nabla \cdot (A \times B) &=& [\epsilon\_{ijk} A\_j B\_k],\_{i} \\
 &=& \epsilon\_{ijk} A\_{j\_{,i}} B\_k + \epsilon\_{ijk} A\_j B\_{k\_{,i}} \\
 &=& B\_k ( \epsilon\_{kij} A\_{j\_{,i}}) - A\_j ( \epsilon\_{jik} B\_{k\_{,i}}) \\
 &=& B \cdot (\nabla \times A) - A \cdot ( \nabla \times B )
\end{eqnarray\*}

is it possible to rewrite it using supreindex (contravariant vector/tensor components) and subindex (covariant ones) ?

Could be it starts with something like this:

\begin{eqnarray\*}
 \nabla \cdot (A \times B) &=& [\epsilon^i\_{jk} A^j B^k]\_{,i} \\
 &=& \epsilon^i\_{jk} A^j\_{,i} B^k + \epsilon^i\_{jk} A^j B^k\_{,i} \\
\end{eqnarray\*}

but I do not known how to continue.