- The Standard Model gauge group is not a subgroup of G2 (complexified or other). Reductive groups have a "rank", and the Standard Model gauge group has rank 4, while G2 has rank only 2, which makes it impossible for the former to be a subgroup of the latter. On the other hand, the Standard Model gauge group is a subgroup of SU(5) in a very beautiful way. I explained that here:
https://arxiv.org/abs/0904.1556
Octonions are wonderful, but obviously you don't want to use them *directly* to model anything that has a associative multiplication. More typically you build things using the octonions, and then consider symmetry groups of those things: these groups will of course have an associative multiplication. That's what I'm doing in the paper you were commenting on.
