On ideals of free Lie algebras (Q1094519)

It is proved that in a free Lie algebra a nontrivial ideal which has infinite rank may be the whole algebra under certain conditions. It is proved that if F is a free Lie algebra of rank \(> 1\) and L a subalgebra of F such that \(L\supseteq F_{2,2}\) then \(L/F_{2,2}\) is free metabelian if and only if there is a set \(Y\subseteq L\) which is linearly independent modulo \(F_ 2\) and \(<Y>+F_{2,2}=L\), where \(| Y| >1\). It is then established that if Y is a subset of F satisfying \(| Y| >1\), linearly independent modulo \(F_ 2\) and if \(L=<Y>+F_{2,2}\) is an ideal of F then \(L=F\).