the transpose map on $M_n(D)$ and $\tau(X)$ signifies applying $\tau$ to the entries of $X$. It is well-known that every central simple algebra with involution $(A,{\sigma})$ is of the form $(M_n(D),\operatorname{ad}_h)$, where $n$ is unique, $D$ is unique up to isomorphism and $h$ is unique up to multiplicative equivalence (see [@BOI 4.A]).

If ${\sigma}$ is of the first kind, then ${\sigma}$ is called *orthogonal* or *symplectic* if ${\sigma}$ becomes adjoint to a quadratic or alternating form, respectively, after scalar extension to a *splitting field* of $A$ (i.e., an extension field $L$ of $K$ such that $A{\otimes}_K L\cong M_n(L)$). We denote the subspace of ${\sigma}$-symmetric elements of $A$ by $\operatorname{Sym}(A,{\sigma})$.

Let $\leq$ be an ordering on $F$. We identify $\leq$ with its *positive cone* $P=\{x\in F\,|\, 0\leq x\}$ via $$x\leq y {\Leftrightarrow}y-x\in P$$ for all $x,y\in F$. In this case we also write $\leq_P$ instead of $\leq$.

Procesi and Schacher [@PS §1] consider central simple algebras $A$, equipped with a *positive involution* ${\sigma}$, i.e., an involution whose *involution trace form* $T_{\sigma}$ is positive semidefinite with respect to the ordering $\leq_P$ on $F$, $$T_{\sigma}(x):=\operatorname{Trd}(\sigma(x)x)\geq_P 0 \quad\text{for all } x\in A.$$ Here $\operatorname{Trd}:A\to F$ (the *trace*) denotes the reduced trace $\operatorname{Trd}_{A/F}$ if ${\sigma}$ 