Let $k$ be a field, and $A$ an associative $k$-algebra with an identity element. Say that $A$ is *quadratic* if any subalgebra of $A$ generated by a single element has dimension at most two.


I am looking for a reference for the following (or any similar) result:



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> Assume that ${\rm char}(k)\ne2$, and let $A$ be a quadratic division algebra over $k$ with $\dim\_k(A)\ge3$. Then $A$ is a quaternion algebra.
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(I expect this to be well known, but am not aware of any reference.)



Charaterisation of quaternion algebras
Yes, this is a great and well-studied question with a super nice answer! See Theorem 3.5.1 of my book (<http://quatalg.org>). You can also say something in characteristic 2 (see Theorem 6.2.8) if you refine "quadratic" = "degree 2" to "nonidentity standard involution".


