On the lower central series quotients of a graded associative algebra

We continue the study of the lower central series Li(A) and its successive quotients Bi(A) of a noncommutative associative algebra A, defined by L1(A)=A, Li+1(A)=[A,Li(A)], and Bi(A)=Li(A)/Li+1(A). We describe B2(A) for A a quotient of the free algebra on two or three generators by the two-sided ideal generated by a generic homogeneous element. We prove that it is isomorphic to a certain quotient of Kähler differentials on the non-smooth variety associated to the abelianization of A.