Using octooctonions (i.e., octonions with octonion coefficients O×O), this paper expresses select findings from nonassociative quantum theory in harmonized notation: Nonrelativistic and relativistic spin operators, Pauli and Dirac matrices, Dirac equation with electromagnetic and gravitational field, and dimensional reduction from quaternionic spin. A generalization of the dimensional reduction program is proposed to argue that octooctonion algebra is wide enough to model a speculated quantum theory that contains all symmetries of the Standard Model, together with four-dimensional Euclidean quantum gravity. The most narrow candidate for such a formulation consists of four generalized Dirac matrices and a four-dimensional operator space with associated fields and charges. Algebraic properties of this relation will be discussed, together with a landscape choice between all possible octooctonionic relations of similar kind.
Jens Köplinger. "Nonassociative quantum theory on octooctonion algebra." J. Phys. Math. 1 1 - 14, December 2009. https://doi.org/10.4303/jpm/S090501