Bott type periodicity for the higher octonions

We study the series of complex nonassociative algebras On and real nonassociative algebras $O_{p,q}$ introduced in [10]. These algebras generalize the classical algebras of octonions and Clifford algebras. The algebras $O_{n}$ and $O_{p,q}$ with $p + q = n$ have a natural $Z_n^2$-grading, and they are characterized by cubic forms over the field $Z_2$. We establish a periodicity for the algebras $O_{n}$ and $O_{p,q}$ similar to that of the Clifford algebras Cln and $Cl_{p,q}$.