  In this article we describe varieties of Lie algebras via algebraic
exponentiation, a concept introduced by Gray in his Ph.D. thesis. For
$\mathbb{K}$ an infinite field of characteristic different from $2$, we prove
that the variety of Lie algebras over $\mathbb{K}$ is the only variety of
non-associative $\mathbb{K}$-algebras which is a non-abelian locally
algebraically cartesian closed (LACC) category. More generally, a variety of
$n$-algebras $\mathcal{V}$ is a non-abelian (LACC) category if and only if
$n=2$ and $\mathcal{V}=\mathsf{Lie}_\mathbb{K}$. In characteristic $2$ the
situation is similar, but here we have to treat the identities $xx=0$ and
$xy=-yx$ separately, since each of them gives rise to a variety of
non-associative $\mathbb{K}$-algebras which is a non-abelian (LACC) category.
