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.