Exercise 9, § 11

Assume that $\mathfrak{g}$ is semi-simple. Let N (resp. P) be the set of nilpotent (resp. principal nilpotent) elements of $\mathfrak{g}$.

a) Show that P is an open subset of N in the Zariski topology (use Exerc. 6). b) Assume that $\mathfrak{g}$ is splittable. Show that P is dense in N. (Use Prop. 10 and Cor. 2 of Th. 1 of §10.)