Exercise 5, § 5

Let $g$ be a Lie algebra and $(\mathscr{D}^0 g, \mathscr{D}^1 g, \ldots, \mathscr{D}^n g)$ the sequence of derived algebras of $g$ ($n \geq 0,\ \mathscr{D}^{n-1}g \neq \mathscr{D}^n g$). Then $\dim \mathscr{D}^i g / \mathscr{D}^{i+1} g \geq 2^{i-1} + 1$ for $1 \leq i \leq n-2$. (Taking quotients by $\mathscr{D}^n g$, reduce it to the case where $g$ is solvable. Then use the fact that $\mathscr{D}g$ is nilpotent and Exercise 7 (c) of § 4.)