$N_H(a_{33})$ is a sum of four squares in $F$. Thus, by our assumption there are $s_1, s_2, s_3 \in \sum F^{{\times}2}$ such that $$XY=Y s_1+ Xs_2 + s_3.$$ We can now finish the proof with an appeal to Lemma \[lem3.1\], as in the proof of Theorem \[thm3.2\].

By tensoring $(M_3(F), \operatorname{ad}_q)$ with Hamilton’s quaternion division algebra, equipped with a *unitary* involution one obtains a counterexample in the non-split unitary case. We leave the details, which are similar to those in the proof of Theorem \[thm3.3\], to the diligent reader.

From a real algebra perspective it is clear that these counterexamples to the PS Conjecture can easily be seen to work over any formally real field $F$ that admits a proper semiordering (see [@PD §5] for details and unexplained terminology). Given such a field $F$, endowed with a proper semiordering, take negative $a,b\in F$ such that $ab$ is negative as well. Then $q=\langle a,b,ab\rangle$ does not weakly represent $1$ (the quadratic module generated by $\{-a,-b,-ab\}$ is proper) and thus in $M_3(F)$, endowed with the involution $\sigma=\operatorname{ad}_q$, the element $ab$ is totally $\sigma$-positive, but not a sum of hermitian squares (as the proof of Theorem \[thm3.2\] shows).

Positive Results {#para.pos.rel}
================

Procesi 