What is the structure group of Hopf map $\pi ：S^{15}\rightarrow S^8$?

Question

What is the structure group of Hopf map $\pi ：S^{15}\rightarrow S^8$?

I know that $\pi ：S^{15}\rightarrow S^8$ is a fiber bundle with fiber $S^7$. However $S^7$ cannot be a Lie group. But every fiber bundle has the structure group, then what is the structure group of $\pi ：S^{15}\rightarrow S^8$?

Answer 1

A priori it's $\text{Diff}(S^7)$. You might hope that it's possible to reduce it to something smaller like $O(8)$, which I think follows from the octonionic construction.