Let $k$ be an algebraic number field. I understand that given a finite set of non-complex places $S\subset V(k)$ of even cardinality, there exists a unique quaternion algebra $Q$ over $k$ such that $Q$ ramifies at all $v\in S$ (i.e. $Q \otimes_k k_v$ is a division algebra) and $Q$ splits at all the other places. Is there a way to get an explicit description of $Q$ of the form $Q(a,b|k)$ for some parameters $a,b, \in k^{\times}$?

The reason for my question is to find an explicit example of a division quaternion algebra over the field $\mathbb{Q}(\sqrt[4]{2})$ that splits at all the archimedean places and is moreover of the form $Q_0 \otimes_{\mathbb{Q}(\sqrt{2})} \mathbb{Q}(\sqrt[4]{2})$ for a quaternion algebra $Q_0$ defined over $\mathbb{Q}(\sqrt{2})$.

Any hint on the general question or on the specific example is greatly appreciated. Thank you!