Exercise 5, § 4

(a) Show that a Lie algebra $g$ in which every subalgebra is subinvariant is nilpotent. (Show that if $\dim g > 1$ every $x \in g$ belongs to an ideal $h \neq g$ of $g$ which has the same property as $g$ and hence is nilpotent by induction on the dimension of $g$; hence $\mathrm{ad}_x x$ is nilpotent.) (b) Show that if in a Lie algebra $g$ every subalgebra distinct from $g$ is distinct from its normalizer, $g$ is nilpotent. (Reduce it to (a).)