A classical problem in nonassociative algebras involves the existence of simple finite-dimensional commutative nilalgebras. In this paper, we study the class Omega of nonassociative algebras satisfying the identity (xy)(2) = x(2)y(2) over a field of characteristic different from 2 and 3. We show that every unitary algebra in Omega is associative. Next, we prove that each prime algebra in Omega is either associative or its center vanishes. For nilalgebras, we obtain that every nilalgebra in Omega is an Engel algebra. Finally, we show that every commutative nilalgebra in Omega of nilindex 4 over a field of characteristic not 2, 3 and 5 is solvable of index <= 3.