$Q$. This situation does not occur in the symplectic case, in view of Proposition \[deg4symp.prop\] below.

The symplectic case
-------------------

Symplectic involutions on $A$ are classified up to conjugation by a relative invariant $\Delta$ with values in the Galois cohomology group $H^3(F,\mu_2)$, see[^1] [@KMRT (16.9)]. We denote by $[A]$ the Brauer class of $A$, viewed as an element of $H^2(F,\mu_2)$, and for $\lambda\in F^\times$ we denote by $(\lambda)$ the square class of $\lambda$, viewed as an element in $H^1(F,\mu_2)$. Using the invariant $\Delta$, we show:

\[deg4symp.prop\] Let $\sigma$ be a symplectic involution on $A$. If $A$ contains $Q$ and $(A,\sigma)_{F_Q}$ is hyperbolic, then $(A,\sigma)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$.

Since $A$ contains $Q$, it decomposes as $A=Q\otimes Q'$ for some quaternion algebra $Q'=(a',b')_F$. Let $\tau'$ be an orthogonal involution on $Q'$ of discriminant $a'$. The involution $\tau={\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}}\otimes\tau'$ on $A=Q\otimes Q'$ is of symplectic type, and clearly hyperbolic over $F_Q$ since $(A,\tau)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$. Similarly, for every invertible $y\in\operatorname{Sym}(Q',\tau')$, the involution $$\tau_y=\operatorname{Int}(1\otimes y)\circ\tau={\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}}\otimes(\operatorname{Int}(y)\circ\tau)$$ is such that $(A,\tau_y)$ contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$. We prove below that $\sigma$ is conjugate to $\tau_y$ for a suitable $y$. By [@KMRT (16.18)], the relative discriminant $\Delta_\tau(\tau')$ is given by $$\Delta_\tau(\tau')=(\operatorname{Nrp}_\tau(1\otimes y))\cup [A]\in H^3(F,\mu_2),$$ where $\operatorname{Nrp}_\tau$ is the Pfaffian norm, as defined 