(Here $m=n^2$ if the involution is of the first kind and $m=2n^2$ otherwise.) Given that $Z$ is the field of fractions of the symmetric centre $Z_0$ of $\operatorname{GM}_n({\mathbb{K}},\operatorname{J})$, we may even assume $\alpha_i\in
Z_0$. We also diagonalise $\operatorname{Trd}(x^*ax)$ as $\langle \beta_1,\ldots, \beta_{m}\rangle$ with $\beta_i\in Z_0$. Clearly, $\alpha_i>0$ and one of the $\beta_i$, say $\beta_1$, is negative with respect to the given $*$-ordering $\leq$. Let ${\overline}Z^{ {\rm rc} }$ denote the real closure of $Z$ with respect to this ordering and form $A:=\operatorname{UD}_n({\mathbb{K}},\operatorname{J})\otimes_Z {\overline}Z^{ {\rm rc} }$ endowed with the involution $\sigma=*\otimes {\rm id}$. Then $A$ is a central simple algebra over a real closed (if $\operatorname{J}=$ orthogonal) or algebraically closed field (if $\operatorname{J}=$ unitary). Moreover, its involution $\sigma$ is positive. Hence by the classification result [@PS Theorem 1.2] of Procesi and Schacher, $A$ is either $M_n({\overline}Z^{ {\rm rc} })$ endowed with the transpose (if $\operatorname{J}=$ orthogonal) or $M_n({\overline}Z)$ endowed with the complex conjugate transpose involution (if $\operatorname{J}=$ unitary). Here ${\overline}Z$ is the algebraic closure ${\overline}Z^{ {\rm rc} }(\sqrt{-1})$ of ${\overline}Z^{ {\rm rc} }$ and the complex conjugate maps $r+t \sqrt{-1} 
\mapsto r- t \sqrt{-1}$ for $r,t\in{\overline}Z^{ {\rm rc} }$.

For $b\in\operatorname{GM}_n({\mathbb{K}},\operatorname{J})$ let $\hat b\in {\mathbb{K}}{\langle{\overline}X,{\overline}X^*\rangle}$ denote a preimage of $b$ 