Raising and lowering on binary homogeneous polynomials ($P=x{\partial }_y,Q=y{\partial }_x,[P,Q]e_k=(n-2k)e_k$)

On the degree-$n$ homogeneous polynomials use $e_k=x^{n-k}{y}^k$. Then $P{e}_k=k{e}_{k-1}$ and $Q{e}_k=(n-k)e_{k+1}$; both maps are nilpotent of index $n+1$. Their commutator is diagonal in this basis with distinct eigenvalues $n,n-2,\dots ,-n$. Thus the raising and lowering operators are not diagonalizable for $n\ge 1$, while their commutator is.