Generalized vector products, duality, and octonionic identities in D=8 geometry

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.