Exercise 15, § 4

Let $l$ be a Lie algebra with zero centre, $\mathfrak{D}_1$ the Lie algebra of derivations of $l$, $\mathfrak{D}_2$ the Lie algebra of derivations of $\mathfrak{D}_1$, ... Then $l$ is an ideal of $\mathfrak{D}_1$, $\mathfrak{D}_1$ is an ideal of $\mathfrak{D}_2$, etc. (§ 1, Exercise 15 (c)). Let $\mathfrak{D}$ be the Lie algebra of derivations of $\mathscr{C}^\infty l$ and $a$ the centre of $\mathscr{C}^\infty l$.

(a) Then $\dim \mathfrak{D}_1 \leq \dim \mathfrak{D} + \dim a$. (Use Exercise 14 above and Exercise 15 of § 1.)

(b) Deduce from (a) that for $i$ sufficiently large all the derivations of $\mathfrak{D}_i$ are inner.