Generalized vector products, duality, and octonionic identities in <i>D</i>=8 geometry

Abstract

In an explicit, unified, and covariant formulation, we study and generalize the exceptional vector products in R8 of Zvengrowski, Gray, and Kleinfeld. We derive the associated general quadratic, cubic, and quartic G2 invariant algebraic identities and uncover an octonionic counterpart to the d=4 quaternionic duality. When restricted to seven dimensions the latter is an algebraic statement of absolute parallelism on S7. We further link up with the Ogievetski–Tzeitlin vector product and obtain explicit tensor forms of the SO(7) and G2 structure constants. SO(8) transformations related by Triality are characterized by means of invariant tensors. Possible physical applications are discussed.