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I guess that what you want to say is that for $k=0, 1,3, 7$ the sphere $S^k$ is a H-space (as in the comment).

Also, Lie groupgroups are a word to look at for having good'' products (better than the above, this works for $S^k$ with $k= 0,1,3$ not $7$, see the comment below).

See Theorem 2.16 of the following nice book.

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I guess that what you want to say is that for $k=0, 1,3, 7$ the sphere $S^k$ is a Lie group.

See Theorem 2.16 of the following nice book.