Exercise 4, § 4

Assume that $k=\mathbf{C}$ and that G is connected. Prove the equivalence of the following properties:

(i) G is nilpotent.

(ii) Every element $\not= 1$ of G is regular.

(Show first that (ii) implies

(ii)$'$ Every element $\not= 0$ of $\mathfrak{g}$ is regular.

Remark next that, if $\mathfrak{g}\not= 0$, then $\mathfrak{g}$ contains elements $x\not= 0$ such that ad $x$ is nilpotent, cf. §3, Exerc. 10. Deduce that $\mathfrak{g}$ is nilpotent, hence (i).)