(i.e., a Positivstellensatz) is presented in the next subsection.

As in the classical construction of the algebra of generic matrices (see e.g. [@Row §1.3]), it is possible to construct the algebra of generic matrices *with involution* [@PS §II]. To each type of involution (orthogonal, symplectic and unitary) an algebra of generic matrices with involution can be associated, as we now explain. We assume from now on that $K$ is a field of characteristic $0$ with involution $*$ and fixed field $F$.

Let $K{\langle{\overline}X,{\overline}X^*\rangle}$ be the free algebra with involution over $(K,*)$, i.e., the algebra with involution, freely generated by the noncommuting variables ${\overline}X:=
(X_1,X_2,\ldots )$. Its elements (called *NC polynomials*) are (finite) linear combinations of words in (the infinitely many) letters ${\overline}X, {\overline}X^*$.

Fix a type J $\in\{$orthogonal, symplectic, unitary$\}$. Let ${\mathfrak{a}}_{\operatorname{J}_n}\subseteq K{\langle{\overline}X,{\overline}X^*\rangle}$ denote the ideal of all identities satisfied by degree $n$ central simple $K$-algebras with type $\operatorname{J}$ involution. That is, $f=f(X_1,\ldots,X_k,X_1^*,\ldots
,X_k^*)\in K{\langle{\overline}X,{\overline}X^*\rangle}$ is an element of ${\mathfrak{a}}_{\operatorname{J}_n}$ if and only if for every central simple algebra $A$ of degree $n$ with type $\operatorname{J}$ involution $\sigma$ and every $a_1,\ldots,a_k\in A$, $$f(a_1,\ldots,a_k,\sigma(a_1),\ldots,\sigma(a_k))=0.$$ Then $\operatorname{GM}_n(K,{\operatorname{J}}):=
K{\langle{\overline}X,{\overline}X^*\rangle}/{\mathfrak{a}}_{\operatorname{J}_n}$ is the *algebra of generic $n\times n$ matrices with type $\operatorname{J}$ involution*.

An alternative 