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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.

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migrated from mathoverflow.net Apr 27 '18 at 10:10

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    $\begingroup$ This looks like homework. What's the motivation? $\endgroup$ – David White Apr 25 '18 at 0:54
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    $\begingroup$ "such that $\mathbb{R}^3\subset A$" is senseless (or has many possible meanings) $\endgroup$ – YCor Apr 25 '18 at 21:32

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