I would like to know if there are some central-simple algebras $D\_1$, $D\_2$ and $D\_3$ over a field $k$ satisfying the following properties :


* $ind(D\_1)=exp(D\_1)=4$ ($ind$ is the Schur index and $exp$ the exponent);
* $D\_2$ and $D\_3$ are two non-isomorphic quaternion algebras;
* $ind(D\_1^{\otimes2} \otimes D\_2)=ind(D\_1^{\otimes 2} \otimes D\_3)=4$ (recall that $ind(D\_1^{\otimes 2})$ is always $2$ in this setting).


To give a little motivation, i'm interested in the existence of "square roots" (in the Brauer group of a field) of quaternion algebras.



Central division and quaternion algebras
You can find such division algebras over $\mathbb{Q}(x\_1,x\_2,x\_3)$ or $k(x\_1,x\_2,x\_3,x\_4)$ where $k$ is any field using the results of "Nakayama, T. Über die direkte Zerlegung einer Divisionsalgebra. Japanese J. of Mathematics 12 (1935), 65–70". A simplified proof is given in the book "Associative algebras" by Pierce, Corollary c, p. 381.


In the notation of Pierce, let $r=3$, $n\_1 = 4$, $n\_2 = 2$, $n\_3=2$ and set $D\_i$ to be the cyclic algebras denoted as $B\_i$ by Pierce. The only thing one needs to observe is that $B\_1^{\otimes 2}$ is equivalent to the cyclic algebra associated to the quadratic subfield of $E\_1$ (and the element $x\_1$) so we may apply the corollary to $D\_1^{\otimes 2} \otimes D\_2$
and $D\_1^{\otimes 2} \otimes D\_3$.


(Sorry for not giving more details but the theorem is long to write out in full. If anything is not clear please leave a comment.)


