Exercise 14, § 2

Assume that $k$ is of characteristic zero. Let V be a finite dimensional $k$-vector space, $\mathfrak{g}$ a Lie subalgebra of $\mathfrak{g}\mathfrak{l}$(V), $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$ and $\mathfrak{n}_V$ the largest ideal of nilpotency of the $\mathfrak{g}$-module V (Chap. I, §4, no. 3, Def. 2). Show that an element of $\mathfrak{h}$ is nilpotent if and only if it belongs to $\mathfrak{n}_V$. (Reduce to the case where the $\mathfrak{g}$-module V is semi-simple, and use Cor. 3 of Th. 2.)