The descent of biquaternion algebras in characteristic two

In this paper we associate an invariant to a biquaternion algebra $B$ over a field $K$ with a subfield $F$ such that $K\mathrm{/}F$ is a quadratic separable extension and $\mathrm{char}(F)=2$. We show that this invariant is trivial exactly when $B\cong B_0\otimes K$ for some biquaternion algebra $B_0$ over $F$. We also study the behavior of this invariant under certain field extensions and provide several interesting examples.