  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.
