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