An appeal to Corollary \[cor4.3\] finishes the proof.

Let $A=M_n(F)$ be a split algebra of $2$-power degree $n=2^\ell$, equipped with a symplectic involution ${\sigma}$. The PS Conjecture holds for $(A,{\sigma})$.

If ${\sigma}$ is a symplectic involution, it is hyperbolic (since it is adjoint to an alternating form over $F$ which is automatically hyperbolic) and we are finished by Corollary \[cor4.4\].

In fact, the PS Conjecture is true for *any* split algebra with symplectic involution. Such an algebra is always of even degree.

\[thm4.7\] Let $A=M_n(F)$ be a split algebra of even degree $n=2m$, equipped with a symplectic involution ${\sigma}$. The PS Conjecture holds for $(A,{\sigma})$.

Since ${\sigma}$ is symplectic, the quadratic form $T_{\sigma}$ is hyperbolic (see [@Lew p. 227] or [@BOI Proof of 11.7]). Thus $T_{\sigma}\simeq m{\times}{\langle}1,-1{\rangle}$ and it suffices to show that $-1$ is a sum of hermitian squares in $A$. We identify $-1$ with $-I_n$, where $I_n$ denotes the $n{\times}n$ identity matrix in $A=M_n(F)$.

Since ${\sigma}$ is symplectic, we have ${\sigma}=\operatorname{Int}(S)\circ t$, where $t$ denotes transposition and $S\in {\mathrm{GL}}_n(F)$ satisfies $S^t=-S$. Since $S$ is skew-symmetric, there exists a matrix $P\in {\mathrm{GL}}_n(F)$ such that $P^tSP=B$, where $B$ is the block diagonal matrix with $m$ blocks $\left[\begin{smallmatrix}
0&1\\
-1 & 