Reducible complements of Lie algebra radicals (Q1075418)

This paper establishes for finite dimensional Lie algebras of characteristic p a counterpart of Levi's ''radical splitting theorem'' for Lie algebras of characteristic 0. Reducible Lie algebras are Lie algebras whose solvable radicals Solv L are nilpotent. For every Cartan subalgebra H, Solv L has a reducible complement S invariant under ad \(H_ s\), that is, \(L=S+Solv L\) and \(S\cap Solv L\) is nilpotent. The key for choosing S invariant under ad \(H_ s\) is a generalization of Engel's criteria for nilpotence. When L is a Lie p-algebra, S can be chosen such that every torus of L of maximal rank is in the same toral switching class as some torus of S.