Exercise 11, § 1

Let $\mathfrak{h}$ be a nilpotent Lie algebra, V an $\mathfrak{h}$-module not reduced to 0, and $\lambda$ a function on $\mathfrak{h}$ such that $V = V^{\lambda}(\mathfrak{h})$. Prove the equivalence of the following properties:

(i) $\lambda$ is a linear form on $\mathfrak{h}$, zero on $[\mathfrak{h},\mathfrak{h}]$.

(ii) There exists a basis of V with respect to which the endomorphisms defined by the elements of $\mathfrak{h}$ are triangular.

(To prove that (i) $=\Rightarrow$ (ii), apply Engel’s theorem to the $\mathfrak{h}$-module $\mathscr{L}(W_{\lambda},V)$, where W is the 1-dimensional $\mathfrak{h}$-module defined by $\lambda .$)

Properties (i) and (ii) are true if $k$ is of characteristic 0 (Prop. 9).