Cross Product of vectors in 7 Dimensions

Question

While reading a geometry book I came across something like.......
Cross Product is possible only in 3 Dimension system and 7 Dimension system.
Why?(or How?)

Answer 1

You probably won't like the answer, but ... this is a deep result. The first proof I saw of it was in Milnor and Stasheff's Characteristic Classes. To read that, you'll need to know about algebraic topology, at least at the level of Vick's Homology Theory book, which'll require some abstract algebra (Herstein's book?)and to know about manifolds, perhaps at the level of Guillemin and Pollack's Differential Topology book. To read that will require a knowledge of a bit of point-set topology (Munkres book, first few chapters), and a good deal of multivariable calculus.

If I knew a simple explanation, I promise I would give it. :(