Levi decomposition of nilpotent centralisers in classical groups

We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the reductive quotients as stated in [Liebeck, Seitz; Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras].