Exercise 13, § 11

Assume that $\mathfrak{g}$ is semi-simple.

a) Let $(x, h, y)$ be a principal $\mathfrak{s}\mathfrak{l}_2$-triplet in $\mathfrak{g}$, and let $\mathfrak{b}$ be the Borel subalgebra containing $x$ (Exerc. 7). Show that $\mathfrak{b}$ is contained in Im ad $x$.

b) Assume that $k$ is algebraically closed. Show that, for any element $z$ in $\mathfrak{g}$, there exist $x, t\in \mathfrak{g}$, with $x$ principal nilpotent, such that $z= [x, t]$ (apply $a$) to a Borel subalgebra containing z).