Abstract

In this paper, we deal with commutative algebras A satisfying the identity 2 beta{(xy)(2) - x(2)y(2)} + gamma{((xy)x)y + ((xy)y)x - (y(2)x)x - (x(2)y)y} = 0, where beta, gamma are scalars. These algebras appeared as one of the four families of degree four identities in Carini, Hentzel and Piacentini-Cattaneo [2]. We prove that if the algebra A admits an identity element, then A is associative. We also prove that there exist trace forms on A. Finally, we prove that if A has a non-degenerate trace form, then A satisfies the identity ((yx)x)x = y((xx)x), a generalization of right alternativity. Our results require characteristic not equal 2, 3.