Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

A theorem of Albert-Draxl states that if a tensor product of two quaternion division algebras $Q_1$, $Q_2$ over a field $F$ is not a division algebra, then there exists a separable quadratic extension of $F$ that embeds as a subfield in $Q_1$ and in $Q_2$. We establish a modified version of this result where the tensor product of quaternion algebras is replaced by the corestriction of a single quaternion algebra over a separable field extension. As a tool in the proof, we show that if the transfer of a nonsingular quadratic form $\varphi$ over a quadratic extension is isotropic for a linear functional $s$ such that $s(1)=0$, then $\varphi$ contains a nondegenerate subform defined over the base field.