On the derived algebra of a centraliser

Let g be a classical Lie algebra, e∈g a nilpotent element and ge⊂g the centraliser of e. We prove that ge=[ge,ge] if and only if e is rigid. It is also shown that if e∈[ge,ge], then the nilpotent radical of ge coincides with [ge(1),ge], where ge(1)⊂ge is an eigenspace of a characteristic of e corresponding to the eigenvalue 1.