On the lower central quotients of the free groups of finite rank of a certain variety

Let Fn be a free group of rank n, and γk(Fn) the k-th term of the lower central series of Fn. For l⩾1, we denote by the quotient group of Fn by a normal subgroup γ2(γ3(Fn))γl+2(γ2(Fn)). In this paper, we show that each of the graded quotients of the lower central series of the group for any l⩾1 is a free abelian group, and give a basis of it by using a generalized Chen's integration in free groups.