Lemma 3

Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{r}$ its radical, $\varphi$ the canonical homomorphism from $\mathfrak{g}$ to $\mathfrak{g}/\mathfrak{r},v$ an elementary automorphism of $\mathfrak{g}/\mathfrak{r}$. There exists an elementary automorphism $u$ of $\mathfrak{g}$ such that $\varphi \circ u=v\circ \varphi$.

We can assume that $v$ is of the form $e^{adb}$, where $b\in \mathfrak{g}/\mathfrak{r}$ and ad $b$ is nilpotent. Let $\mathfrak{s}$ be a Levi subalgebra of $\mathfrak{g}$ (Chap. I, §6, no. 8, Def. 7) and let $a$ be the element of $\mathfrak{s}$ such that $\varphi (a) =b$. Since ad$_{\mathfrak{s}}a$ is nilpotent, ad$_{\mathfrak{g}}a$ is nilpotent (Chap. I, §6, no. 3, Cor. of Prop. 3), and $u=e^{ad_{\mathfrak{g}}a}$ is an elementary automorphism of $\mathfrak{g}$ such that $\varphi \circ u=v\circ \varphi$.