Exercise 14, § 6

Let $X$ be a nilpotent matrix in $\mathfrak{gl}(n, K)$. Show that there exist in $\mathfrak{gl}(n, K)$ two other matrices $Y, H$ such that $[H, X] = X,\ [H, Y] = -Y,\ [X, Y] = H$. (Argue by induction on $n$, using the Jordan form of $X$; thus reduce it to the case where $X = E_{12} + E_{23} + \cdots + E_{n-1,n}$; then take $Y$ to be a combination of the $E_{k,k-1}$ and $H$ a combination of the $E_{jj'}$.)