Abstract. Minimal non-elementarty finite groups must be nilpotent. The Lie algebra analogue admits non-nilpotent examples. We classify them for complex solvable Lie algebras. In this note we find all such finite dimensional Lie algebras with nilpotent derived algebra over an algebraically closed field.
A groupThe Lie algebras considered here are solvable. Hence the Frattini subalgebra coincides with the Frattini ideal. Let φ(L) denote the Frattini subalgebra of L. For solvable Lie algebras over a field of characteristic 0, the derived algebra is nilpotent; hence the theorem finds all minimal non-elementary Lie algebras in this case when the field is algebraically closed. For characteristic p, if L is nilpotent of length 2, then the final term, L w , in the lower central series is nilpotent, hence abelian. Then L is the semi-direct sum of L w and a Cartan subalgebra C of L [1, Theorem 8]. Then C is abelian and L 2 = L w . Hence the theorem applies to any Lie algebra of nilpotent length 2 over an algebraically closed field.
Lemma. Let L be a minimal non-elementary finite dimensional solvable Lie algebra withProof. (i) Suppose that L is nilpotent. Then φ(H) = H 2 = 0 for every proper subalgebra H of L and φ(L) = L 2 . If dim(L/L 2 ) > 2, then every pair of