A unital multiplication on a positive-dimensional sphere forces odd dimension

Let \(n>0\) and let \(e\in S^n\). Suppose there is a map \[ \mu:S^n\times S^n\longrightarrow S^n \] such that \[ \mu(x,e)=x=\mu(e,x) \] for every \(x\in S^n\). Prove that \(n\) is odd.

(In fact \(n\) must be \(1\), \(3\), or \(7\), but this stronger result is considerably harder.)