The iteration that builds systematic forms for Hadamard matrices doubles the dimension each application equivalent to the Cayley-Dickson algebra doubling process. These systematic forms have a row or column composition rule where the composition of any two rows(columns) is closed for the matrix, such that for sequentially assigned column and row indexes from 0 up, the index of the result is the binary bit-wise exclusive or (xor) of the indexes for the two rows(columns) participating. The Exclusive Or Group of order 2 exponent n and all of its subgroups is isomorphic to the product tables for order 2 exponent n Cayley-Dickson algebra and all of its subalgebras when all resultant –1 signs are dropped. The Boolean xor operator thus forms the bridge between Cayley-Dickson algebras and Hadamard matrices, but only for the division algebra subset. The dimension 8 Hadamard matrix is ubiquitous within Octonion algebraic structure by specifying optimal enumerations of, and relationships between; basis elements, Quaternion subalgebra triplets, proper Octonion orientations, Octonion Algebraic Invariance and Variance.