Exercise 2, § 4

Let $g$ be a semi-direct product of an algebra $b$ of dimension 1 and a commutative ideal $g'$. Let $x \in b, x \neq 0$, and $u$ be the restriction of $\mathrm{ad}_x x$ to $g'$. (a) For $g$ to be nilpotent, it is necessary and sufficient that $u$ be nilpotent. (b) For the Killing form of $g$ to be zero, it is necessary and sufficient that $\mathrm{Tr}(u^2) = 0$. (c) Deduce from (a) and (b) that there exist non-nilpotent Lie algebras whose Killing forms are zero. (d) Deduce from (a) that in a nilpotent Lie algebra such that $\mathscr{C}^{p-1} g \neq \{0\}$, $\mathscr{C}^p g = \{0\}$ it is possible that $\mathscr{C}_i g \neq \mathscr{C}^{p-i} g$.