Exercise 12, § 4

Let g be a Lie algebra and $\mathscr{C}^\infty g$ the intersection of the $\mathscr{C}^g g$. (a) The Lie algebra $g / \mathscr{C}^\infty g$ is nilpotent. (b) Show that there exists a nilpotent subalgebra $h$ of $g$ such that $g = h + \mathscr{C}^\infty g$. (Argue by induction on $\dim g$. Suppose that $g$ is not nilpotent. Let $x \in g$ be such that $I = \bigcap_k (\mathrm{ad}\, x)^k(g) \neq \{0\}$. Let $n$ be the union of the kernels of the $(\mathrm{ad}\, x)^k$. Show that this is a subalgebra of $g$ and that $g$ is the direct sum of I and $n$. By the induction hypothesis $n$ is the sum of $\mathscr{C}^\infty n$ and a nilpotent subalgebra $h$. Finally, $\mathscr{C}^\infty n \subset \mathscr{C}^\infty g$ and $I \subset \mathscr{C}^\infty g$.

(c) Verify that the following multiplication table defines a (solvable) Lie algebra $g$ of dimension 5:

and the other $[x_i, x_j]$ are zero. Show that for this Lie algebra

but that there exists no supplementary subalgebra of $\mathscr{C}^\infty g$ in $g$.