General Cayley–Dickson construction gave a slight generalization, defining the product and involution on for an algebra with involution (with ) to be for an additive map that commutes with and left and right multiplication by any element. (Over the reals all choices of are equivalent to −1, 0 or 1.) In this construction, is an algebra with involution, meaning: is an abelian group under has a product that is left and right distributive over has an involution , with , , . The algebra produced by the Cayley–Dickson construction is also an algebra with involution.
General Cayley–Dickson construction gave a slight generalization, defining the product and involution on for an algebra with involution (with ) to be for an additive map that commutes with and left and right multiplication by any element. (Over the reals all choices of are equivalent to −1, 0 or 1.) In this construction, is an algebra with involution, meaning: is an abelian group under has a product that is left and right distributive over has an involution  with , , . The algebra produced by the Cayley–Dickson construction is also an algebra with involution.