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We study a series of real nonassociative algebras $\mathbb{O}_{p,q}$introduced in $[5]$. These algebras have a natural $\mathbb{Z}_2^n$-grading,where $n=p+q$, and they are characterized by a cubic form over the field$\mathbb{Z}_2$. We establish all the possible isomorphisms between the algebras$\mathbb{O}_{p,q}$ preserving the structure of $\mathbb{Z}_2^n$-graded algebra.The classification table of $\mathbb{O}_{p,q}$ is quite similar to that of thereal Clifford algebras $\mathrm{Cl}_{p,q}$, the main difference is that thealgebras $\mathbb{O}_{n,0}$ and $\mathbb{O}_{0,n}$ are exceptional.

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