Let $K$ be a field and let $Q$ be a quaternion algebra over $K$. Then it is well-known that the class $[Q]$ of $Q$ in $Br(K)$ has order $2$. One can show this by constructing an explicit isomorphism $Q \otimes\_K Q \cong M\_2(K)$. My question is about the converse. 



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> Does there exist a field $K$ and a division algebra $D$ over $K$ such that the class $[D]$ of $D$ in $Br(K)$ has order $2$, but such that $D$ is not isomorphic to a quaternion algebra over $K$?
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If such a $K$ and $D$ exist, then it would also be nice to see an explicit example. 


As a non-example, I believe that it follows from local and global class field theory that if $K$ is a local or global field, then every element of order $2$ in $Br(K)$ may indeed be represented by a quaternion algebra.



Brauer group elements of order $2$
I have a vague memory that examples of $2$-torsion classes which are not exactly quaternion algebras can be found in one of the papers on Merkurjev's Theorem in "Applications of algebraic K-theory to algebraic geometry and number theory, Part 2", Contemporary Math. 1986 (either the one of Merkurjev, or the one of Wadsworth), but I can not check.


On the other hand, there are examples and references in this paper of A. Kresch:


arxiv.org/pdf/math/0009115


