Bott type periodicity for the higher octonions

We study the series of complex nonassociative algebras $\bbO_n$ and real nonassociative algebras $\bbO_{p,q}$ introduced in~\cite{MGO2011}.
These algebras generalize the classical algebras of octonions and Clifford algebras. The algebras $\bbO_{n}$ and $\bbO_{p,q}$ with $p+q=n$ have a natural $\Z_2^n$-grading, and they are
characterized by cubic forms over the field $\Z_2$. We establish a periodicity for the algebras~$\bbO_{n}$ and $\bbO_{p,q}$ similar to that of the Clifford algebras $\mathrm{Cl}_{n}$ and~$\mathrm{Cl}_{p,q}$.