I have heard sometimes that the only dimensions $k$ for which there exists a "*good*" smooth product $P:S^k\times S^k\to S^k$ are $k = 0,1,3,7$ (the above products corresponding to $\mathbb Z\_2, U(1)\subset\mathbb C$, the product of unit quaternions and of unit Cayley numbers).


I would like to ask for references about such a result. 


More precisely, I am interested in finding out how is it that one can define "*good*" so that the above is true (with proofs, preferably).



Are there good product rules on the $k$-sphere?
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 groups 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](http://www.math.cornell.edu/~hatcher/VBKT/VBdoublepage.pdf). 


