  There exists a well known construction which allows to associate with two
hyperbolic affine spheres $f_i: M_i^{n_i} \to \mathbb R^{n_i+1}$ a new
hyperbolic affine sphere immersion of $I \times M_1 \times M_2$ into $\mathbb
R^{n_1+n_2+3}$. In this paper we deal with the inverse problem: how to
determine from properties of the difference tensor whether a given hyperbolic
affine sphere immersion of a manifold $M^n \to \mathbb R^{n+1}$ can be
decomposed in such a way.
