is of the first kind and the composition $\operatorname{Trd}_{K/F}\circ \operatorname{Trd}_{A/K}$ if ${\sigma}$ is of the second kind. The form $T_{\sigma}$ is a nonsingular quadratic form over $F$, cf. [@BOI §11]. If $\dim_K A=n$, then $\dim T_{\sigma}=n$ if ${\sigma}$ is of the first kind and $\dim T_{\sigma}=2n$ if ${\sigma}$ is of the second kind.

The notion of positive involution seems to have been considered first by Weil in his groundbreaking paper [@Wei].

Procesi and Schacher also define a notion of *positive elements* in $(A,{\sigma})$, cf. [@PS §V]. For greater clarity we have adapted their definitions as follows:

1.  An ordering $\leq_P$ of $F$ is called a [*$\sigma$-ordering*]{} if it makes the involution $\sigma$ positive, i.e., if $$0\leq_P \operatorname{Trd}(\sigma(x)x) \quad \text{for all }x\in A.$$

2.  Suppose $\leq_P$ is a $\sigma$-ordering on $F$. An element $a\in \operatorname{Sym}(A,{\sigma})$ is called ${\sigma}$-*positive with respect to $\leq_P$* if the quadratic form $ \operatorname{Trd}(\sigma(x)ax)$ is positive semidefinite with respect to $\leq_P$. That is, if $$0\leq_P\operatorname{Trd}(\sigma(x)ax)\quad \text{for all } x\in A.$$

3.  An element $a\in \operatorname{Sym}(A,{\sigma})$ is called *totally ${\sigma}$-positive* if it is positive with respect to [all]{} $\sigma$-orderings on $F$.

Elements of the form ${\sigma}(x)x$ with $x\in A$ are called *hermitian squares*. The set of hermitian squares 