That is, suppose that a subset S of the octonions $\mathbb{O}$ is a group under octonionic multiplication. Does it follow that S is contained in the Quaternions $\mathbb{H}$?



Are associative "subgroups" of octonions necessarily quaternionic?
Are you assuming that the subset is multiplicatively closed? If not, exactly what do you mean by "associative"? Do you mean associations of length 3, or of all lengths?


If $S$ is associative, then the $\mathbb R$-linearity of multiplication shows that the vector subspace generated by $S$ is associative (simply break it up into terms and use associativity on each one). If you mean all associations, then that generates an associative subalgebra and you're done. (Being contained in a division algebra is enough to prove Frobenius.)


I'm not sure what to do if only associations of length 3 are allowed. There is probably still a proof of some kind.


