algebras with arbitrary involution over $K$ and with common fixed field $F$. The PS Conjecture holds for the tensor product $\bigotimes_{i=1}^\ell (Q_i,{\sigma}_i)$.

This is an immediate consequence of Proposition \[prop4.1\] and Theorem \[thm4.2\].

Let $A=M_n(F)$ be a split algebra of $2$-power degree $n=2^\ell$, equipped with an orthogonal involution ${\sigma}$ which is adjoint to an $n$-fold Pfister form over $F$. The PS Conjecture holds for $(A,{\sigma})$.

By Becher’s proof of the Pfister Factor Conjecture [@Bec], $(A,{\sigma})$ decomposes as $$(A,{\sigma})\cong \bigotimes_{i=1}^\ell (Q_i,{\sigma}_i),$$ where $(Q_1,{\sigma}_1),\ldots, (Q_\ell,{\sigma}_\ell)$ are quaternion algebras with involution. An appeal to Corollary \[cor4.3\] finishes the proof.

\[cor4.4\] Let $A=M_n(K)$ be a split algebra of $2$-power degree $n=2^\ell$, equipped with a hyperbolic involution ${\sigma}$ of any kind. Let $F$ be the fixed field of $(A,{\sigma})$. The PS Conjecture holds for $(A,{\sigma})$.

Recall from [@BST Theorem 2.1] that the involution ${\sigma}$ is hyperbolic if there exists an idempotent $e\in A$ such that ${\sigma}(e)=1-e$ or, equivalently, if the adjoint (quadratic, alternating or hermitian) form of ${\sigma}$ is hyperbolic.

If $\ell=1$ this is just the split version of Proposition \[prop4.1\]. Assume now that $\ell\geq 2$. By [@BST Theorem 2.2], $(A,{\sigma})$ decomposes as $$(A,{\sigma})\cong \bigotimes_{i=1}^\ell (Q,{\sigma}_i),$$ where $Q=M_2(K)$ and ${\sigma}_1,\ldots, {\sigma}_\ell$ are involutions on $Q$. 