Nonzero Octonions as a 7-sphere While reading about Moufang loops in the book "An introduction to Quasigroups and their Representations" by Smith, I've encountered the following statement:

The set $ S^7 $ of nonzero octonions of norm 1 forms a Moufang loop under multiplication. Geometrically, this set is a 7-sphere.

While I understand why this set indeed forms a Moufang loop, I'm not sure how it is viewed as a sphere, or what is the general connection between Moufang loops and this geometrical point of view. Could anyone elaborate?

Mariano's answer is of course right on the money.
Since I myself happen to currently have my library's copy of Conway and Smith's book Quaternions and Octonions, let me recommend this book to you as well.  In general, throughout the book they continually emphasize a geometric approach to the algebraic structures mentioned in the title, so it would be a good place to read up on the topic of your question.  
