Octonions, simple Moufang loops and triality

Abstract

Nonassociative finite simple Moufang loops are exactly the loops constructed by Paige from Zorn vector matrix algebras. We prove this result anew, using geometric loop theory. In order to make the paper accessible to a broader audience, we carefully discuss the connections between composition algebras, simple Moufang loops, simple Moufang 3-nets, $S$-simple groups and groups with triality. Related results on multiplication groups, automorphisms groups and generators of Paige loops are provided.