On the structure and zero divisors of the Cayley-Dickson sedenion algebra

The algebrasC (complex numbers), H (quaternions), andO (octonions) are real division algebras obtained from the real numbers R by a doubling procedure called the Cayley-Dickson Process. By doubling R (dim 1), we obtain C (dim 2); then C produces H (dim 4); and H yields O (dim 8): The next doubling process applied to O then yields an algebra S (dim 16) called the sedenions. This study deals with the subalgebra structure of the sedenion algebra S and its zero divisors. In particular, it shows that S has subalgebras isomorphic to R; C; H; O; and a newly identifled algebra e O called the quasi-octonions that contains the zero-divisors of S: