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.