  We define a special matrix multiplication among a special subset of $2N\x 2N$
matrices, and study the resulting (non-associative) algebras and their
subalgebras. We derive the conditions under which these algebras become
alternative non-associative and when they become associative. In particular,
these algebras yield special matrix representations of octonions and complex
numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix
representation of octonions also yields elegant insights into Dirac's equation
for a free particle. A few other results and remarks arise as byproducts.
