On the derived algebra of a centraliser

Let $\g$ be a classical Lie algebra, $e$ a nilpotent of $\g$ element and $\gt g_e$ the centraliser of $e$ in $\g$. We prove that $\g_e=[\g_e,\g_e]$ if and only if $e$ is rigid. It is also shown that if $e$ is contained in $[\g_e,\g_e]$, then the nilpotent radical of $\g_e$ coincides with $[\g(1)_e,\g_e]$, where $\g(1)_e$ is an eigenspace of a characteristic of $e$ with the eigenvalue 1.