It is known that for any compact Lie group $G$ with maximal torus $T$, that any other maximal torus $T'$ is conjugate to $T$. This might be a bit of a stretch, but I was wondering if it is possible to use this result to deduce which positive dimensional spheres can have a group structure (or maybe just a Lie group structure). The program I had in mind was this:

1) Show that if $S^n$ has a Lie group structure that its maximal torus would be $S^1$

2) Look at the set of all maximal tori (each one being a copy of $S^1$) and somehow turn each of these into the fiber of some bundle $S^n\rightarrow B$.

3) Hopefully be able to conclude that $n$ must be $1$ or $3$ (my idea was that perhaps we could show that $B= S^{n-1}$

Any suggestions would be greatly appreciated. Also, I understand that this is using much more machinery than necessary to attack the problem of which spheres admit group structures but the idea still makes me curious if there is something like this.

Edit: I don't think you need this much to deduce which spheres admit a Lie group structure; I'm not sure if this argument provides a path to showing which spheres admit a topological group structure.