Let $A$ be a (non-commutative, not nessasarilly associative) division algebra over $\mathbb{R}$ such that $\mathbb{R}^3 \subset A$. Assume that for any two nonzero vectors $u, v \in \mathbb{R}^3$ we have the following property: if $\overset{\sim} u, \overset{\sim} v$ are obtained from $u, v$ by a rotation in a 2-plane containing $u, v$, then $\overset{\sim} u (\overset{\sim} v)^{-1}=uv^{-1}$. Show that $A \neq \mathbb{R}^3$. Classify all such algebras $A$.
I was told that this problem originally lead Hamilton to the invention of quaternions though I was not able to find any such recollection in his papers.