On Restricting Subsets of Bases in Relatively Free Groups

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety A_mA_n, and let A = {a_1,…, a_r} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {a_{l+1},…,a_r}, then S is a subset of a basis for the relatively free group on {a_1,…,a_l}.