and let $A$ be a quaternion algebra over $K$ with unitary involution ${\sigma}$ whose restriction to $K$ is $\tau$, where $\tau$ is determined by $\tau(\sqrt{\delta})=-\sqrt{\delta}$. By a well-known result of Albert [@BOI 2.22] there exists a unique quaternion $F$-subalgebra $A_0\subseteq A$ such that $$A=A_0{\otimes}_F K \text{ and } \sigma= \gamma_0{\otimes}\tau,$$ where $\gamma_0$ is quaternion conjugation on $A_0$. Then $T_{\sigma}\simeq T_{\gamma_0}{\otimes}T_\tau\simeq T_{\gamma_0}{\otimes}{\langle}1,-\delta{\rangle}$. Since $\tau(\sqrt{\delta})\sqrt{\delta}=-\delta$, we are finished by the symplectic part of the proof.

This shows in particular that the PS Conjecture is true for full matrix algebras of degree two over a formally real field $F$ since these are just split quaternion algebras.

Part (ii) of the proof of Proposition \[prop4.1\] motivates the following more general result:

\[thm4.2\] Let $A$ and $B$ be central simple algebras with centre $K$, equipped with arbitrary involutions ${\sigma}$ and $\tau$, respectively. Assume that $(A,{\sigma})$ and $(B,\tau)$ have the same fixed field $F$. If the PS Conjecture holds for $(A,{\sigma})$ and $(B,\tau)$, it also holds for the tensor product $(A{\otimes}_K B, {\sigma}{\otimes}\tau)$.

This is a simple computation, using the fact that $T_{{\sigma}{\otimes}\tau}\simeq T_{\sigma}{\otimes}T_\tau$ and that elements of $A$ commute with elements of $B$ in the tensor product $A{\otimes}_K B$.

\[cor4.3\] Let $(Q_1,{\sigma}_1),\ldots, (Q_\ell,{\sigma}_\ell)$ be quaternion 