An Algebraic Proof Sedenions Are Not A Division Algebra And Other Consequences Of Cayley dickson Algebra Definition Variation
The Cayley Dickson dimension doubling algorithm nicely maps R C H O S and beyond, but without consideration of any possible definition variation. Quaternion Algebra H has two orientations, and they drive definition variation in all subsequent algebras, all of which have H as a subalgebra. Requiring Octonion Algebra O to be a normed composition algebra limits the possible orientation combinations of its seven H subalgebras to sixteen proper O orientations, which are itemized. Identification of the O subalgebras for Sedenion Algebra S and orientation limitations on these subalgebras provides a fully algebraic proof that all O subalgebras cannot be oriented as proper Octonion Algebras, verifying Sedenion Algebra is not generally a normed composition division algebra. A simple mnemonic form for validating proper O orientations is provided. The method to partition any number of O algebraic element products into product term sets with like responses to all possible proper O orientation changes; either to a single algebraic invariant set or to one of 15 different algebraic variant sets, is provided. Most important for O based mathematical physics, the stated Law of Octonion Algebraic Invariance requires observables to be algebraic invariants. Its converse, The Law of the Unobservable suggests homogeneous equations of algebraic constraint built from the algebraic variant sets. These equations of constraint are important to mathematical physics since they will limit the family of solutions for the differential equations describing reality and do not have their genesis in experimental observation. An alternative to the Cayley Dickson doubling scheme which builds by variations is provided.