Embedding of a Lie algebra into Lie-admissible algebras

Abstract

Let A be a flexible Lie-admissible algebra over a field of characteristic ≠ \ne 2, 3. Let S be a finite-dimensional classical Lie subalgebra of A − {A^ - } which is complemented by an ideal R of A − {A^ - } . It is shown that S is a Lie algebra under the multiplication in A and is an ideal of A if and only if S contains a classical Cartan subalgebra H which is nil in A and such that H H ⊆ S HH \subseteq S and [ H , R ] = 0 [H,R] = 0 . In this case, the multiplication between S and R is determined by linear functionals on R which vanish on [R, R]. If A is finite-dimensional and of characteristic 0 then this can be applied to give a condition that a Levi-factor S of A − {A^ - } be embedded as an ideal into A and to determine the multiplication between S and the solvable radical of A − {A^ - } .