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 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. 

