Exercise 1, § 4

Let $g$ be a nilpotent Lie algebra and $p$ (resp. $q$) the smallest integer such that $\mathscr{C}^p g = \{0\}$ (resp. $\mathscr{C}_q g = g$). Show that $p = q + 1$ and that $\mathscr{C}_i g \supset \mathscr{C}^{p-i} g$. (Use the argument of Proposition 1.)