Criteria for the existence of a Jordan–Chevalley–Seligman decomposition

Let (L,[p])(L,[p]) be a finite dimensional restricted Lie algebra over a field KK of positive characteristic p  . A Jordan–Chevalley–Seligman decomposition of x∈Lx∈L is a unique expression of x as a sum of commuting semisimple and nilpotent elements in L  . It is well-known that each x∈Lx∈L has such a decomposition when KK is perfect. When KK is non-perfect, the present paper gives several criteria for the existence of a Jordan–Chevalley–Seligman decomposition for a given x∈Lx∈L as well as for determining when an element in the restricted subalgebra generated by x is semisimple or nilpotent.