Proposition 2

Let V be a finite dimensional $\mathfrak{g}$-module and $s\in$ Aut$_0(\mathfrak{g})$.

(i) There exists $S\in \mathbf{G}\mathbf{L}(V)$ such that $(s(x))_V= Sx_VS^{-1}$ for all $x\in \mathfrak{g}$.

(ii) If $s\in$ Aut$_e(\mathfrak{g})$, S can be chosen to be an element of $\mathbf{S}\mathbf{L}(V)$ leaving stable all the $\mathfrak{g}$-submodules of V.

Assertion (ii) follows from Lemma 1 (i). Now let $s\in$ Aut$_0(\mathfrak{g})$ and denote by $\rho$ the representation of $\mathfrak{g}$ defined by V. By (ii), the representations $\rho$ and $\rho \circ s$ become equivalent after extension of scalars. They are therefore equivalent (Chap. I, §3, no. 8, Prop. 13), hence the existence of S.