In a previous paper , the quotient groups of the lower central series Ḡn = Gn/Gn+1 were studied. There the group G was assumed to be a free product of a finite number of finitely generated Abelian groups and Gn denoted the nth subgroup of the lower central series of G. Here we give an improved proof of a complicated lemma which first appeared in  (in particular, Lemma 4.4 of ). The proof given here, especially for property (iii) of the conclusion of that lemma, is a significant simplification of that which appears in . We observe that one of the consequences of Lemma 4.4 of  is to give a set of free generators for the lower central quotients in the case where the free factors are torsion free (i.e., G = J in the terminology of . Moreover the free generators are the J-basic commutators also using the terminology of Definition 4.1 of ). The authors only thought of this simplification after the publication of . Furthermore, our improved proof uses results from  and .
In this paper, we will employ the notation, terminology, results, references, and equations of . Furthermore, the numbering, a.b, of any definition, equation, etc., of  will correspond here to a.b-I. For example, Lemma 2.1-I will mean Lemma 2.1 of .
In order to carry out our goal, we need to give some preliminary machinery.