Classification of the algebras $\mathbb{O}_{p,q}$

We study a series of real nonassociative algebras $O_{p,q}$ introduced in [5]. These algebras
have a natural $Z^n_2$-grading, where $n = p + q$, and they are characterized by a cubic form
over the field $Z_2$. We establish all the possible isomorphisms between the algebras $O_{p,q}$
preserving the structure of $Z^n_2$-graded algebra. The classification table of $O_{p,q}$ is quite
similar to that of the real Clifford algebras $Cl_{p,q}$, the main difference is that the algebras
$O_{n,0}$ and $O_{0,n}$ are exceptional.