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 $x\in V$ 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.
Categories: math.RT