A characterisation of Lie algebras via algebraic exponentiation

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 $\mathcal{V}={\mathsf{L}\mathsf{i}\mathsf{e}}_{\mathbb{K}}$. 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.