and Schacher [@PS p. 404 and 405] prove their conjecture for central simple algebras $A$ of degree two, i.e., quaternion algebras, with arbitrary involution ${\sigma}$ by appealing to matrices and the Cayley–Hamilton theorem. We start this section by giving an alternative argument motivating some of the generalizations that follow.

Throughout this section we assume that the base field $F$ is formally real.

\[prop4.1\] Let $A$ be a quaternion algebra not necessarily division with centre $K$, equipped with an arbitrary involution ${\sigma}$. Let $F$ be the fixed field of $(A,{\sigma})$. Each entry occurring in a diagonalisation of $T_{\sigma}$ is a sum of hermitian squares.

\(i) We first consider involutions of the first kind on $A$. Let $A$ be the quaternion algebra $(a,b)_F$ with $F$-basis $\{1,i,j,k\}$ where $i$, $j$ and $k$ anti-commute, $ij=k$, $i^2=a$ and $j^2=b$.

If ${\sigma}$ is symplectic, then ${\sigma}$ is the unique quaternion conjugation involution $\gamma$ on $A$. An easy computation gives $T_{\sigma}=T_\gamma\simeq {\langle}2{\rangle}{\otimes}{\langle}1,-a,-b,ab{\rangle}$. We have $$1=\gamma(1)1,\ -a=\gamma(i)i,\ -b=\gamma(j)j,\ ab=\gamma(k)k.$$

If ${\sigma}$ is orthogonal, then ${\sigma}=\operatorname{Int}(u)\circ\gamma$, where $u\in A$ satisfies $\gamma(u)=-u$. From [@BOI 11.6] we know that $$T_{\sigma}\simeq {\langle}2{\rangle}{\otimes}{\langle}1, \operatorname{Nrd}_A(u), -\operatorname{Nrd}_A(s), -\operatorname{Nrd}_A(su){\rangle}$$ for some $s\in A$ with ${\sigma}(s)=s=-\gamma(s)$. Now, $$\begin{aligned}
\operatorname{Nrd}_A(u)&=u\gamma(u) =u\gamma(u) u^{-1}u={\sigma}(u)u;\\
-\operatorname{Nrd}_A(s)&=-\gamma(s)s={\sigma}(s)s;\\
-\operatorname{Nrd}_A(su)&=-\operatorname{Nrd}_A(s)\operatorname{Nrd}_A(u)=-\gamma(s)s\operatorname{Nrd}_A(u)={\sigma}(s){\sigma}(u)us={\sigma}(us)us.\end{aligned}$$

\(ii) Finally, let $K=F(\sqrt{\delta})$ 