If $S^a$ is a sphere which acts transitively on itself, then $a\in \{0,1,3\}$.

Question

I'm reading an article in which the author says that a sphere $S^a$ which acts transitively on itself must be $S^0$, $S^1$ or $S^3$. And I really cannot see why.

Answer 1

Actually, any sphere that is a Lie group at all must be a 0, 1, or 3-sphere. For a proof, see Is there an easy way to show which spheres can be Lie groups?