Brauer group elements of order $2$ 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. 

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 $K$?

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.
 A: Yes. Such an example is provided by a biquaternion algebra over $K$. It is a well known result due to Albert (see Lam, Introduction to Quadratic Forms over Fields) that the tensor product of two quaternion algebras is a division algebra iff they do not have a common quadratic splitting field.
A: I think what you're after is, in modern parlance, an algebra of period 2 but index strictly greater than 2.  Googling "period-index problem" gives plenty of references and examples.
I had a look on Colliot-Thélène's web page and found the following article: "Exposant et indice d'algèbres centrales simples non ramifiées" (avec  un appendice par Ofer Gabber),  L'Enseignement Mathématique 48 (2002) 127–146.  In that article, there is a nice introduction with a big list of references.  Apparently the first example of the type you're after was given by Brauer himself:  R. Brauer, "Untersuchungen über die arithmetischen Eigenschaften von Gruppen linearer Substitutionen", Zweite Mitteilung, Math. Zeitschrift 31 (1929) 733–747.  The article by Brauer is available online, but only one page at a time from a rather clunky web site, so I haven't gone into it to track down the example.
A: By the Merkurjev-Suslin theorem, $K^M_2(K) \to \mathrm{Br}(K)[2]$ is surjective in characteristic $\neq 2$, and $K^M_2(K)$ is generated by symbols $\{a,b\}$, $a,b\in K^\times$.  For characteristic $2$, see Gille-Szamuely, Chapter 9.
A: 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
