Octonions, Albert vectors and the group ${}^2 \mathrm{E}_6(F)$

Abstract

We extend the explicit octonionic approach to groups of type $\mathrm{E}_6$ to the twisted groups ${}^2\mathrm{SE}_{6,K}(F)$ associated with a quadratic Galois extension $K/F$. These groups act on the Albert space over $K$, preserving the Dickson--Freudenthal determinant and a Hermitean form. When the field norm $K^{\times}\to F^{\times}$ is surjective, we give explicit generators and coördinate proofs of the three-orbit theorem on white points and of the stabilisers of representative white vectors. Under this hypothesis we also prove simplicity of the central quotient. Over finite fields we recover the classical orbit lengths and group orders from the stabiliser orders and the total number of white points. Finally, without assuming norm surjectivity, we show that the two isotropic white-point orbits remain unchanged, while the non-isotropic orbits are parametrised by $F^{\times}/\mathrm{N}_{K/F}(K^{\times})$.