$\DeclareMathOperator\Mat{Mat}$How to construct an explicit isomorphism of the split quaternion algebra $(a,b)_F$ over the field $F$ to $\Mat_2(F)$?

As it is known that the algebra of quaternions is either a division ring or isomorphic to $\Mat_2(F)$ and in the second situation this quaternions are called split, I am interested in finding an algorithm to construct this explicit isomorphism.

I saw some result for $F = \mathbb{Q}(\sqrt{d})$ by Kutas, but what is the case with other fields, is there any not necessarily polynomial algorithm for them?