Exercise 13, § 5

Let $g$ be a Lie Algebra and $\mathfrak{D}$ the Lie algebra of derivations of $g$. For $g$ to be characteristically nilpotent, it is necessary and sufficient that $\mathfrak{D}$ be nilpotent and that $\dim g > 1$. (To see that the condition is sufficient, write $g$ as a sum of subspaces $g^\lambda$, applying Exercise 12 to the identity mapping of $\mathfrak{D}$. Show that $[g^\lambda, g^\mu] \subset g^{\lambda+\mu}$ and that each $g^\lambda$ is an ideal of $g$. Deduce that $g^\lambda$ is commutative for $\lambda \neq 0$. Using again the fact that $\mathfrak{D}$ is nilpotent, show that $g = g^0$ if $\dim g > 1$. Otherwise $g = g^0 \times h$, where $h$ is commutative. Show first that $\dim h \leq 1$. If $\dim h = 1$, note that there would exist a derivation D of $g$ such that $D(g^0) = \{0\}$ and $D(h)$ is contained in the centre of $g^0$ (§ 4, Exercise 8 (a)).)