Exercise 6, § 11

Assume that $\mathfrak g$ is absolutely simple, and denote its Coxeter number by $h$. Let $x$ be a nilpotent element of $\mathfrak g$. Show that $(\operatorname{ad}x)^{2h-1}=0$ and that

if and only if $x$ is principal. (Reduce to the case in which $\mathfrak g$ is split, and $x$ is contained in the subalgebra $\mathfrak n_+$ of Prop. 10. Repeat the proof of Prop. 10.)