Fox-type problems in enveloping algebras

Let L be a Lie algebra with the universal enveloping algebra U(L). The augmentation ideal ω(L) of U(L) is the associative ideal of U(L) generated by L. Let S be a subalgebra of L and n a positive integer. In this paper we prove that L∩ω2(L)ωn(S)=γn+2(S)+γn+1(S∩γ2(L)), where γn+2(S) is the (n+2)-th term of the lower central series of S.