We give classifications of group gradings, up to equivalence and up to isomorphism, on the tensor product of a Cayley algebra $\mathcal{ C}$ and a Hurwitz algebra over a field of characteristic different from 2. We also prove that the automorphism group schemes of $\mathcal{ C}^{ \otimes n}$ and $\mathcal{ C}^n$ are isomorphic. On the other hand, we prove that the automorphism group schemes of a Smirnov algebra (a $35$-dimensional simple exceptional structurable algebra constructed from a Cayley algebra $\mathcal{ C}$) and $\mathcal{ C}$ are isomorphic. This is used to obtain classifications, up to equivalence and up to isomorphism, of the group gradings on Smirnov algebras.
Categories: math.RA