Exercise 8, § 4

Let $g$ be a Lie algebra, $m$ an ideal of codimension 1, $x$ an element of $g$ not belonging to $m$ and $z \in g$.

(a) Show that the linear mapping $D$ which reduce to 0 on $m$ and maps $x$ to $z$ is a derivation if $z$ belongs to the centralizer $a$ of $m$ in $g$.

(b) Let $q$ be the largest integer such that $a \subset \mathscr{C}^q g$. Show that if further $z \notin \mathscr{C}^{q+1} g$, $D$ is not an inner derivation of $g$.

(c) Deduce from (a) and (b) and Exercise 7 (c) that if $g$ is a nilpotent Lie algebra of dimension $> 1$, the vector space of inner derivations of $g$ is of co-dimension $\geq 2$ in the space of all derivations of $g$.