Algebras with involution that become hyperbolic over the fonction field of a conic

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We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra Q. We classify these algebras in degree 4 and give an example of such a division algebra with orthogonal involution of degree 8 that does not contain Q with its canonical involution, even though it contains Q and is totally decomposable into a tensor product of quaternion algebras with involution.