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.

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