Matrix representation of octonions and generalizations

We define a special matrix multiplication among a special subset of 2Nx2N m atrices, and study the resulting (nonassociative) algebras and their subalg ebras. We derive the conditions under which these algebras become alternati ve nonassociative, and when they become associative. In particular, these a lgebras yield special matrix representations of octonions and complex numbe rs; they naturally lead to the Cayley-Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac's equat ion for a free particle. A few other results and remarks arise as byproduct s. (C) 1999 American Institute of Physics. [S0022-2488(99)03108-4].