Let F denote a commutative ring, \(F\) the corresponding ring of polynomials in two non-commuting indeterminates, and F[X,Y] the ring of polynomials in two commuting indeterminates. A polynomial \(f(X,Y)\in F\) is called admissible if each of its monomials has length at least 3 and f(X,Y) has trivial image under the natural F-algebra map from \(F\) to F[X,Y]. In general, the F-algebras R studied in the paper need not be commutative. The purpose of this paper is to continue the study of commutativity of these rings. Conditions imposed on R to obtain commutativity are variations of the following result: Let R be a ring; and suppose that for each x,y\(\in R\), there exists a polynomial p(X)\(\in XZ[X]\), depending on x and y, for which \(xy-yx=(xy-yx)p(x)\). Then R is commutative.