Rational orbits of the space of pairs of exceptional Jordan algebras

Abstract

Let $k$ be a field of characteristic not equal to $2,3$, $\mathbb{O}$ an octonion over $k$ and $\mathcal{J}$ the exceptional Jordan algebra defined by $\mathbb{O}$. We consider the prehomogeneous vector space $(G,V)$ where $G=GE_6\times \mathrm{GL}(2)$ and $V=\mathcal{J}\oplus \mathcal{J}$. We prove that generic rational orbits of this prehomogeneous vector space are in bijective correspondence with $k$-isomorphism classes of pairs $(\mathcal{M},\mathfrak{n})$ where $\mathcal{M}$'s are isotopes of $\mathcal{J}$ and $\mathfrak{n}$'s are cubic \'etale subalgebras of $\mathcal{M}$. Also we prove that if $\mathbb{O}$ is split, then generic rational orbits are in bijective correspondence with isomorphism classes of separable extensions of $k$ of degrees up to $3$.