Exercise 3, § 4

(a) Let $g$ be a nilpotent Lie algebra, $z$ its centre and $b$ a non-zero ideal of $g$. Show that $z \cap b \neq \{0\}$. (Consider $b$ as a $g$-module by means of the adjoint representation.) (b) If in a Lie algebra $g$ an ideal $b$ is contained in $\mathscr{C}_{i+1} g$ but not in $\mathscr{C}_i g$, show that the ideals $b \cap \mathscr{C}_k g$ are all distinct for $0 \leq k \leq i + 1$. (Apply (a).)