Clifford algebra-parametrized octonions and generalizations

Introducing products between multivectors of Cl0,7 (the Clifford algebra over the metric vector space R) 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 S, and the XY -product. This generalization is accomplished in the uand (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 ⊕ R) × Cl0,7 → R ⊕ R , Cl0,7 × (R ⊕ R ) → R ⊕ R and Cl0,7 × Cl0,7 → R ⊕ R . 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 Oalgebras 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 ⊕ R →֒ 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.