Exercise 13, § 4

Let $g$ be a Lie algebra and $\mathfrak{z}$ the centralizer of $\mathscr{C}^\infty g$ in $g$. If $\mathfrak{z} \not\subset \mathscr{C}^\infty g$, $g$ has non-zero centre. (Write $g = \mathscr{C}^\infty g + \mathfrak{t}$, where $\mathfrak{t}$ is a nilpotent subalgebra of $g$ (Exercise 12). Let $g_1 = \mathfrak{z} + \mathfrak{t}$. Let $\mathfrak{h}$ be a nilpotent subalgebra such that $g_1 = \mathscr{C}^\infty g_1 + \mathfrak{h}$. Show that $g = \mathscr{C}^\infty g + \mathfrak{h}$ and that $\mathscr{C}^\infty g_1 \subset \mathfrak{z} \cap \mathscr{C}^\infty g$. Let $y$ be an element of $\mathfrak{z}$ not belonging to $\mathscr{C}^\infty g$. Write $y = x + x'$ where $x \in \mathfrak{h}, x' \in \mathscr{C}^\infty g_1$; then $x \in \mathfrak{z}$ and $x \neq 0$ and hence $\mathfrak{z} \cap \mathfrak{h}$ is a non-zero ideal of $\mathfrak{h}$; using Exercise 3 (a), deduce that there exists a non-zero element of $g$ permutable with $\mathscr{C}^\infty g$ and $\mathfrak{h}$.)