Alternative algebras with the hyperbolic property

We investigate the structure of an alternative finite dimensional $\text{Q}$-algebra $\mathfrak{A}$ subject to the condition that for a $\mathbb{Z}$-order $\mathrm{\Gamma }\subset \mathfrak{A}$, and thus for every $\mathbb{Z}$-order of $\mathfrak{A}$, the loop of units of $\text{U}(\mathrm{\Gamma })$ does not contain a free abelian subgroup of rank two. In particular, we prove that the radical of such an algebra associates with the whole algebra. We also classify $RA$-loops $L$ for which $\mathbb{Z}L$ has this property. The classification for group rings is still an open problem.