Did Hamilton have a proof that $\mathbb{R}^3$ is cannot be turned into an $\mathbb{R}$-division algebra?

Question

It is well-known that $\mathbb{R}^n$ cannot be made into a non-commutative $\mathbb{R}$-division algebra if $n\ne 4$.

My question is whether there is a (slick) proof of this for $n=3$; in particular, I wonder if Hamilton had a proof of this result for $n=3$ which made him look at $n=4$ for discovering quaternions.