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The Weyl groups of the fine gradings on the exceptional Lie algebra e 6 whose universal grading group is infinite are computed.

We give classifications of group gradings, up to equivalence and up to isomorphism, on the tensor product of a Cayley algebra C and a Hurwitz algebra over a field of characteristic different from 2. We also prove that the automorphism group schemes of C ⊗n and C n are isomorphic.On the other hand, we prove that the automorphism group schemes of a Smirnov algebra T(C) (a 35-dimensional simple exceptional structurable algebra constructed from a Cayley algebra C) and C are isomorphic. This is used to obtain classifications, up to equivalence and up to isomorphism, of the group gradings on Smirnov algebras. ⋆ Supported by the Spanish Ministerio de Economía y Competitividad-Fondo Europeo de Desarrollo Regional (FEDER) MTM2017-83506-C2-1-P. A.S. Córdova-Martínez also acknowledges support from the Consejo Nacional de Ciencia y Tecnología (CONACyT, México) through grant 420842/262964.

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