Clifford algebra-parametrized octonions and generalizations

Introducing products between multivectors of Cℓ0,7 (the Clifford algebra over the metric vector space R0,7) and octonions, resulting in an octonion, and leading to the non-associative standard octonionic product in a particular case, we generalize the octonionic X-product, associated with the transformation rules for bosonic and fermionic fields on the tangent bundle over the 7-sphere S7, and the XY-product. This generalization is accomplished in the u- and (u,v)-products, where u,v∈Cℓ0,7 are fixed, but arbitrary. Moreover, we extend these original products in order to encompass the most general—non-associative—products (R⊕R0,7)×Cℓ0,7→R⊕R0,7, Cℓ0,7×(R⊕R0,7)→R⊕R0,7 and Cℓ0,7×Cℓ0,7→R⊕R0,7. We also present the formalism necessary to construct Clifford algebra-parametrized octonions, which provides the structure to present the O1,u algebra. Finally we introduce a method to construct O-algebras endowed with the (u,v)-product from O-algebras endowed with the u-product. These algebras are called O-like algebras and their octonionic units are parametrized by arbitrary Clifford multivectors. When u is restricted to the underlying paravector space R⊕R0,7↪Cℓ0,7 of the octonion algebra O, these algebras are shown to be isomorphic. The products between Clifford multivectors and octonions, leading to an octonion, are shown to share graded-associative, supersymmetric properties. We also investigate the generalization of Moufang identities, for each one of the products introduced.