This paper is dedicated to the memory of Zbigniew Oziewicz, to his
generosity, intelligence and intensity in the search that is science and
mathematics. The paper begins with a basic construction that produces Clifford
algebras inductively, starting with a base algebra A that is associative and
has an involution. This construction is an analog of the Cayley-Dickson
Construction that produces the complex numbers, quaternions and octonions
starting from the real numbers. Our basic construction always produces
associative algebras and can be iterated an indefinite number of times. We
generalize the basic construction to a group theoretic construction where a
group G acts on the algebra A, and show how this group theoretic construction
is related to matrix algebras. The paper then concentrates on applications of
this algebra to the Dirac Equation and the Majorana-Dirac Equation.
