Rational orbits in some prehomogeneous vector spaces associated to $S{p}_6$ revisited

Abstract

Let $k$ be a field with $\text{char}(k)\ne 2$. We prove that all maximal flags of composition algebras over $k$, appear as the $k$-rational $S{p}_6$-orbits in a Zariski-dense $S{p}_6$-invariant subset $V^{ss}\subset V={\wedge }^3{V}_6$, where $V_6$ is the standard $6$-dimensional irreducible representation of $S{p}_6$. This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces $(S{p}_6\times G{L}_1^2,V)$ and $(GS{p}_6\times G{L}_1^2,V)$. We also get all reduced Freudenthal algebras of dimensions $6$ and $9$, represented by the same orbit spaces.