On equivariant maps related to the space of pairs of exceptional Jordan algebras

Let $mathcal{J}$ be the exceptional Jordan algebra and $V=mathcal{J}oplus mathcal{J}$. We construct an equivariant map from $V$ to $mathrm{Hom}_k(mathcal{J}otimes mathcal{J},mathcal{J})$ defined by homogeneous polynomials of degree $8$ such that if $xin V$ is a generic point, then the image of $x$ is the structure constant of the isotope of $mathcal{J}$ corresponding to $x$. We also give an alternative way to define the isotope corresponding to a generic point of $mathcal{J}$ by an equivariant map from $mathcal{J}$ to the space of trilinear forms.