  Let $G$ be a classical group of dimension $d$ and let
$\boldsymbol{a}=(a_1,\dots,a_d)$ be differential indeterminates over a
differential field $F$ of characteristic zero with algebraically closed field
of constants $C$. Further let $A(\boldsymbol{a})$ be a generic element in the
Lie algebra $\mathfrak{g}(F\langle \boldsymbol{a} \rangle)$ of $G$ obtained
from parametrizing a basis of $\mathfrak{g}$ with the indeterminates
$\boldsymbol{a}$. It is known (cf. work by Juan) that the differential Galois
group of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $F\langle
\boldsymbol{a} \rangle$ is $G(C)$. In this paper we construct a differential
field extension $\mathcal{L}$ of $F\langle \boldsymbol{a} \rangle$ such that
the field of constants of $\mathcal{L}$ is $C$, the differential Galois group
of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $\mathcal{L}$ is
still the full group $G(C)$ and $A(\boldsymbol{a})$ is gauge equivalent over
$\mathcal{L}$ to a matrix in normal form which we introduced in work by Seiss.
We also consider specializations of the coefficients of $A(\boldsymbol{a})$.
