How to Obtain Division Algebras from Twisted Cayley–Dickson Doublings

Abstract

We change the order of the factors in the classical Cayley–Dickson doubling process and investigate the eight-dimensional algebras obtained when doubling a quaternion algebra using these new multiplications, allowing the element c used in the doubling process to be an invertible element in the quaternion algebra. By changing the place of c inside the multiplication we obtain different families of algebras and conditions when these algebras are division.