algebra with involution $(M_n(F),\operatorname{ad}_q)$.

Recall that a central simple $F$-algebra with involution $(A,\sigma)$ is hyperbolic if and only if it is isomorphic to $(\operatorname{End}_DV,\operatorname{ad}_h)$ for some hyperbolic hermitian or skew-hermitian form $h$ on a vector space $V$ over a division algebra $D$. In particular, $A$ admits a hyperbolic involution if and only if the index of $A$ divides $\frac12\deg(A)$. If so, then $A$ admits up to conjugation a unique orthogonal hyperbolic involution, and a unique symplectic one; see [@BST] or [@KMRT 6.B].

For any central simple $F$-algebra with involution $(A,\sigma)$ and any field extension $K/F$, we let $(A,\sigma)_K=(A\otimes_FK,\sigma\otimes{\operatorname{Id}})$. As pointed out in the introduction, we shall be mostly interested in the special case where $K=F_Q$ is the function field of the conic associated to a quaternion algebra $Q$, i.e. its Severi–Brauer variety. Throughout the paper, we fix the notation $$Q=(a,b)_F$$ with $a$, $b\in F^\times$, and we assume $Q$ is not split. We may then identify $F_Q$ with the quadratic extension $F(\sqrt{at^2+b})$ of $F(t)$, where $t$ is an indeterminate. For any central simple $F$-algebra $A$, we let $A(t)=A\otimes_FF(t)$ and $A[t]=A\otimes_FF[t]$.

Let $(A,\sigma)$ be a central simple $F$-algebra with involution. Assume $\sigma$ is anisotropic. The following conditions are equivalent:

1.  $(A,\sigma)_{F_Q}$ is 