Remark 2

Let V be a finite dimensional vector space. For all $g\in \mathbf{S}\mathbf{L}$(V), let $\varphi (g)$ be the automorphism $x \rightarrow gxg^{-1}$ of $\mathfrak{g}\mathfrak{l}(V)$. Then

Aut$_e(\mathfrak{g}\mathfrak{l}(V)) =\varphi (\mathbf{S}\mathbf{L}(V))$.

Indeed, by (1), Aut$_e(\mathfrak{g}\mathfrak{l}(V))$ is contained in $\varphi (\mathbf{S}\mathbf{L}$(V)), and the opposite inclusion follows from Algebra, Chap. III, §8, no. 9, Prop. 17 and (1). An analogous argument shows that Aut$_e(\mathfrak{s}\mathfrak{l}(V))$ is the set of restrictions of elements of $\varphi (\mathbf{S}\mathbf{L}(V))$ to $\mathfrak{s}\mathfrak{l}(V)$.