Exercise 3, § 5

Let $g$ be a solvable Lie algebra such that the conditions $x \in g,\ y \in g,\ [[x, y], y] = 0$ imply $[x, y] = 0$. Show that $g$ is commutative. (Let $k$ be the largest integer such that $\mathscr{D}^{k-1}g \neq \{0\},\ \mathscr{D}^k g = \{0\}$. Assuming $k \geq 2$, show first that $[\mathscr{D}^{k-2}g, \mathscr{D}^{k-1}g] = \{0\}$ and then that $[\mathscr{D}^{k-2}g, \mathscr{D}^{k-2}g] = \{0\}$, whence a contradiction.)