Exercise 9, § 5

Let $g$ be a Lie algebra, $r$ its radical, $n$ its largest nilpotent ideal and $D$ a derivation of $g$. Show that $\mathscr{D}g \cap r \subset n$. (Let $d$ be the Lie algebra of derivations of $g$ and $r'$ its radical. If $x \in g$ is such that $Dx \in r$, then

(Corollary 2 to Proposition 5), hence $\mathrm{ad}(Dx)$ is nilpotent (Theorem 1) and hence $Dx \in n$.)