Normal Forms in Differential Galois Theory for the Classical Groups

Let $G$ be a classical group of dimension $d$ and let $\mathit{a}=(a_1,\dots ,a_d)$ be differential indeterminates over a differential field $F$ of characteristic zero with algebraically closed field of constants $C$. Further let $A(\mathit{a})$ be a generic element in the Lie algebra $\mathfrak{g}(F⟨\mathit{a}⟩)$ of $G$ obtained from parametrizing a basis of $\mathfrak{g}$ with the indeterminates $\mathit{a}$. It is known (cf. work by Juan) that the differential Galois group of ${\mathit{y}}^{\mathrm{\prime }}=A(\mathit{a})\mathit{y}$ over $F\langle \boldsymbol{a} \rangle$is$G©$. In this paper we construct a differential field extension $\mathcal{L}$ of $F⟨\mathit{a}⟩$ such that the field of constants of $\mathcal{L}$ is $C$, the differential Galois group of ${\mathit{y}}^{\mathrm{\prime }}=A(\mathit{a})\mathit{y}$ over $\mathcal{L}$ is still the full group $G(C)$ and $A(\mathit{a})$ 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 $A(\mathit{a})$.