Exercise 7, § 11

Assume that $\mathfrak g$ is semi-simple. Let $x$ be a nilpotent element of $\mathfrak g$. Then $x$ is principal if and only if $x$ is contained in a unique Borel subalgebra of $\mathfrak g$. (Reduce to the case in which $k$ is algebraically closed. Use Prop. 10 and Prop. 10 of §3, no. 3.)