Exercise 12, § 1

Let $g$ be a Lie algebra. (a) For a derivation $D$ of $g$ to commute with all the inner derivations of $g$, it is necessary and sufficient that $D$ map $g$ into its centre. (b) Suppose henceforth that $g$ has centre zero, so that $g$ is identified with an ideal of the Lie algebra $\mathfrak{D}$ of all derivations of $g$. Show that the centralizer of $g$ in $\mathfrak{D}$ is zero (use (a)). In particular, the centre of $\mathfrak{D}$ is zero. (c) Show that a derivation $\Delta$ of $\mathfrak{D}$ such that $\Delta(g) = \{0\}$ is zero. ($[ \Delta(\mathfrak{D}), g ] \subset \Delta([\mathfrak{D}, g]) + [\mathfrak{D}, \Delta(g)] \subset \Delta(g) = \{0\}$ and hence $\Delta(\mathfrak{D}) = \{0\}$ by (b).) (d) Show that, if further $g = \mathscr{D}g$, every derivation $\Delta$ of $\mathfrak{D}$ is inner (use (c) and Exercise 11).