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.