Exercise 7, § 4

(a) Let $V$ be a finite-dimensional vector space over $K$, $g$ a Lie algebra of nilpotent endomorphisms of $V$ and $(V_r)_{0 \leq r < n}$ a decreasing sequence of vector subspaces of $V$, with $V_0 = V, V_n = \{0\}$ and $g(V_r) \subset V_{r+1}$ for $0 \leq r < n$. Show by induction on $i$ that $(\mathscr{D}^i g)(V_r) \subset V_{r+2^i}$. If $\dim V \geq 2^i$, the subspace of elements of $V$ annihilated by $\mathscr{D}^i g$ is of dimension $\geq 2^i$. If $\dim V \leq 2^i$, $\mathscr{D}^i g = \{0\}$.

(b) Let $g$ be a nilpotent Lie algebra and $i$ an integer $\geq 0$. If $\dim \mathscr{D}^i g > 2^i + 1$ the centre of $\mathscr{D}^i g$ is of dimension $\geq 2^i$. If $\dim \mathscr{D}^i g \leq 2^i + 1$, $\mathscr{D}^i g$ is commutative. (Apply (a) to the restrictions to $\mathscr{D}^i g$ of the ad$_g$ $x, x \in g$.)

(c) Let $g$ be a nilpotent Lie algebra and $i$ an integer $\geq 0$. If $\mathscr{D}^i g$ is not commutative, $\mathscr{D}^i g/\mathscr{D}^{i+1} g$ is of dimension $\geq 2^i + 1$. (Reduce it, by passing to the quotient, to the case where $\dim \mathscr{D}^{i+1} g = 1$ and use (b).)