Subalgebras of free restricted Lie algebras

Abstract

A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.