It was brought to my attention that in previous papers I put in the public domain (references [1] through [8]), I did a rather piecemeal presentation of the critical concepts of algebraic orientation, algebraic invariance and algebraic variance, as well as the generalization of Cayley-Dickson algebras I referred to as Cayley-Dickson without emphasizing the more general nature of the presentation, which was done for reader familiarity purposes. I will attempt to remedy this within this document starting at the beginning: the basic definition of an algebra, carried through to a full definition and discussion of the general family of hypercomplex algebras I call ℋ which subsumes all division algebras and traditional Cayley-Dickson doubled forms. Taking into account all structural and algebraic orientation options exposes beautiful structure revealed through group theoretical aspects of ℋ construction. The concepts presented are essential to Octonion Algebra mathematical physics.