Exercise 15, § 6

(a) Let $X, Y$ be two matrices of $\mathfrak{gl}(n, K)$. Show that, if $[X, Y] = X$, $X$ is nilpotent. (For every polynomial $f \in K[T]$, observe that

(b) Let $g$ be a Lie algebra and $h, x$ two elements of $g$ such that $[h, x] = x$ and there exists $z \in g$ with $[x, z] = h$. Show that there exists $y \in g$ such that $[x, y] = h$ and $[h, y] = -y$. (Observe first that $[z, h] + z$ belongs to the centralizer $n$ of $x$ in $g$. Then show that $n$ is stable under $\mathrm{ad}\ h$ and that the restriction to $n$ of $I - \mathrm{ad}\ h$ is bijective; to do this, note that $\mathrm{ad}\ x$ is nilpotent, using (a), and prove that, if $g_i$ is the image of $g$ under $(\mathrm{ad}\ x)^i$, $I - \mathrm{ad}\ h$ gives on taking quotients a bijection of $(n \cap g_{i-1})/(n \cap g_i)$; for this use the relation (c) Let $g$ be a subalgebra of $\mathfrak{gl}(n, K)$; suppose that, for every nilpotent matrix $X \in g$, there exist two other matrices $H, Y$ in $g$ such that $[H, X] = X,\ [H, Y] = -Y,\ [X, Y] = H$. Let $b$ be a subalgebra of $g$ such that there exists a subspace $m$ supplementary to $h$ in $g$ and such that $[h, m] \subset m$. Show that $h$ has the same property as $g$ for all its nilpotent matrices (use (b)).