Nonzero Octonions as a 7-sphere

Question

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?

Answer 1

The real octonions are a normed division algebra, and in such a thing you always have a unit sphere. The unit sphere, because of multiplicativity of the norm, is always closed under the product.

The title of your question, though, refers to the nonzero octonions, and that is surly not a sphere (it has the homotopy type of a sphere, and in fact the unit sphere is a strong deformation retract---but I guess that if you know what this term means then you also are aware of this fact!)

Answer 2

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.