Exercise 4, § 4

Let $g$ be a nilpotent Lie algebra. (a) Every subalgebra of $g$ is subinvariant. (Use Proposition 3.) (b) Let $b$ be a vector subspace of $g$ such that $b + \mathscr{D}g = g$. Show that the subalgebra of $g$ generated by $b$ is $g$. (Apply (a) to this subalgebra. Reduce it thus to the case where $b$ is an ideal of $g$ and use Proposition 4 of § 1.) Deduce that the minimum number of generators of $g$ is $\dim g / \mathscr{D}g$.