Remark 1

Let V be a finite dimensional vector space, $\mathfrak{g}$ a Lie subalgebra of $\mathfrak{a}=\mathfrak{g}\mathfrak{l}$(V), $x$ an element of $\mathfrak{g}$ such that ad$_{\mathfrak{g}}x$ is nilpotent. Then there exists a nilpotent element $n$ of $\mathfrak{a}$ such that ad$_{\mathfrak{a}}n$ extends ad$_{\mathfrak{g}}x$. Indeed, let $s, n$ be the semi-simple and nilpotent components of $x$; then ad$_{\mathfrak{a}}s$ and ad$_{\mathfrak{a}}n$ are the semi-simple and nilpotent components of ad$_{\mathfrak{a}}x$ (Chap. I, §5, no. 4, Lemma 2), so ad$_{\mathfrak{a}}s$ and ad$_{\mathfrak{a}}n$ leave $\mathfrak{g}$ stable, and ad$_{\mathfrak{a}}s|\mathfrak{g}$ and ad$_{\mathfrak{a}}n|\mathfrak{g}$ are the semi-simple and nilpotent components of ad$_{\mathfrak{g}}x$; consequently, ad$_{\mathfrak{g}}x=$ ad$_{\mathfrak{a}}n|\mathfrak{g}$, which proves our assertion. In view of Lemma 1 (ii), every elementary automorphism of $\mathfrak{g}$ extends to an automorphism of $\mathfrak{a}$ of the form $u \rightarrow mum^{-1}$ where $m\in \mathbf{G}\mathbf{L}(V)$.