  We consider non-degenerate graph immersions into affine space $\mathbb
A^{n+1}$ whose cubic form is parallel with respect to the Levi-Civita
connection of the affine metric. There exists a correspondence between such
graph immersions and pairs $(J,\gamma)$, where $J$ is an $n$-dimensional real
Jordan algebra and $\gamma$ is a non-degenerate trace form on $J$. Every graph
immersion with parallel cubic form can be extended to an affine complete
symmetric space covering the maximal connected component of zero in the set of
quasi-regular elements in the algebra $J$. It is an improper affine hypersphere
if and only if the corresponding Jordan algebra is nilpotent. In this case it
is an affine complete, Euclidean complete graph immersion, with a polynomial as
globally defining function. We classify all such hyperspheres up to dimension
5. As a special case we describe a connection between Cayley hypersurfaces and
polynomial quotient algebras. Our algebraic approach can be used to study also
other classes of hypersurfaces with parallel cubic form.
