description of the algebra of generic matrices with involution can be obtained as follows. Let $\zeta:=(\zeta_{ij}^{(\ell)}\mid1\leq i, j\leq n,\,\ell\in{\mathbb{N}})$ denote commuting variables and form the polynomial algebra $K[\zeta]$ endowed with the involution extending $*$ and fixing $\zeta_{ij}^{(\ell)}$ pointwise. Consider the $n\times n$ matrices $Y_{\ell}:=\big[
\zeta_{ij}^{(\ell)}\big]_{1\leq i,j\leq n}\in M_n(K[\zeta])$, $\ell\in{\mathbb{N}}$. Each $Y_{\ell}$ is called a *generic matrix*.

1.  If $\operatorname{J}\in\{$orthogonal, unitary$\}$, then the (unital) $K$-subalgebra of $M_n(K[\zeta])$ generated by the $Y_\ell$ and their transposes is (canonically) isomorphic to $\operatorname{GM}_n(K,{\rm J})$.

2.  If $\operatorname{J}=$ symplectic, then $n$ is even, say $n=2m$. Consider the usual symplectic involution $$\begin{bmatrix}
    x&y\\
    z&w
    \end{bmatrix}
    \mapsto
    \begin{bmatrix}
    w^t&-y^t\\
    -z^t&x^t
    \end{bmatrix}$$ on $M_{2m}(K[\zeta])$. Then the (unital) $K$-subalgebra of $M_n(K[\zeta])$ generated by the $Y_\ell$ and their images under this involution is (canonically) isomorphic to $\operatorname{GM}_n(K,{\operatorname{J}})$.

If $n=1$, then $\operatorname{J}\in\{$orthogonal, unitary$\}$ and $\operatorname{GM}_1(K,\operatorname{J})$ is isomorphic to $K[\zeta]$ endowed with the involution introduced above. Hence in the sequel we will always assume $n\geq 2$.

Let $\operatorname{J}\in\{$orthogonal, symplectic, unitary$\}$. For $n\geq 2$, $\operatorname{GM}_n(K,\operatorname{J})$ is a PI algebra and a domain (cf. [@PS §II]). Hence its central localization is a division algebra $\operatorname{UD}_n(K,\operatorname{J})$ with involution, which we call the *generic division algebra* with type $\operatorname{J}$ involution of degree $n$. As we will only 