Some finite varieties of Lie algebras (Q1178054)

A class of Lie algebras is called \(n\)-recognizable if each Lie algebra \(L\) is in the class whenever all \(n\)-generated subalgebras of \(L\) are in the class. All Lie algebras considered are finite-dimensional. Let \(F\) be an algebraically closed field of characteristic \(>5\) or a field of characteristic zero. The authors prove that, over \(F\), solvable Lie algebras, Lie algebras which have a nilpotent ideal with abelian quotient and supersolvable Lie algebras are 2-recognizable. However, Lie algebras over an infinite field which have an abelian ideal with nilpotent quotient are 3-recognizable.