Section \[sec:PSc\] we fix terminology and summarize some of the Procesi–Schacher results in a modern language. Then in Section \[sec:counter\] we present counterexamples to the Procesi–Schacher conjecture, while Section \[para.pos.rel\] contains a study of examples (mainly in the split case) where the conjecture is true.

For general background on central simple algebras with involution we refer the reader to [@BOI] and for the theory of quadratic forms over fields we refer to [@Lam].

The Procesi–Schacher Conjecture {#sec:PSc}
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Let $F$ be a formally real field and let $A$ be a central simple algebra with involution ${\sigma}$ and centre $K$. Assume that $F$ is the fixed field of ${\sigma}$ (i.e., ${\sigma}|_F=\mathrm{id}_F$). The involution ${\sigma}$ is of the *first kind* if $K=F$, and of the *second kind* (also called *unitary*) otherwise. In this case $[K:F]=2$ and ${\sigma}|_K$ is the non-trivial element in $\mathrm{Gal}(K/F)$.

Let $D$ be a division algebra over $K$ with involution $\tau$ and fixed field $F$. Let $h$ be an $n$-dimensional hermitian or skew-hermitian form over $(D,\tau)$. Then $h$ gives rise to an involution on $M_n(D)$, the *adjoint involution* $\operatorname{ad}_h$, defined by $$\operatorname{ad}_h(X)= H\cdot \tau(X)^t\cdot H^{-1},$$ for all $X\in M_n(D)$, where $H$ is the Gram matrix of $h$, $t$ denotes 