On special triples of algebras over the rational function field

Let [Formula: see text] be a field, [Formula: see text], [Formula: see text] nonsquares. Let [Formula: see text] be a biquaternion division algebra over [Formula: see text] such that [Formula: see text], and [Formula: see text] the biquaternion algebra over the rational field [Formula: see text], which is Brauer equivalent to [Formula: see text]. We prove that the algebras [Formula: see text] and [Formula: see text] have no common quaternion subalgebra over [Formula: see text]. In particular, one can choose a field [Formula: see text] with [Formula: see text], so that [Formula: see text]. As a consequence we get that if a cubic polynomial [Formula: see text] is irreducible, [Formula: see text], [Formula: see text], and [Formula: see text] is still a division algebra, then the algebras [Formula: see text] and [Formula: see text] do not have a common slot. Given a biquaternion division algebra [Formula: see text] over [Formula: see text], we also obtain an equivalent condition in terms of Chow groups for the equality [Formula: see text].