Exercise 12, § 6

Let $g$ be a Lie algebra, $r$ its radical and $D$ a derivation of $g$. (a) If $D(r) = \{0\}$, $D$ is inner. (Let $c$ be the centralizer of $r$ in $g$. The adjoint representation of $g$ defines a representation $x^* \mapsto \rho(x^*)$ of $g^* = g/r$ on $c$. On the other hand, $[D(g), r] \subset D([g, r]) + [g, D(r)] = \{0\}$, hence $D(g) \subset c$ and hence $D$ defines a linear mapping $D^*: g^* \to c$. Show that

for all $x^*, y^*$ in $g^*$. Deduce that there exists $a \in c$ such that $D^*x^* = \rho(x^*)a$ for all $x^* \in g^*$ (cf. no. 2, Remark 2), whence $Dx = [x, a]$ for all $x \in g$.) (b) If $D$ coincides on $r$ with an inner derivation of $g$, $D$ is inner. (Apply (a).)