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Let $G$ be a classical group of dimension $d$ and let$\boldsymbol{a}=(a_1,\dots,a_d)$ be differential indeterminates over adifferential field $F$ of characteristic zero with algebraically closed fieldof constants $C$. Further let $A(\boldsymbol{a})$ be a generic element in theLie algebra $\mathfrak{g}(F\langle \boldsymbol{a} \rangle)$ of $G$ obtainedfrom parametrizing a basis of $\mathfrak{g}$ with the indeterminates$\boldsymbol{a}$. It is known (cf. work by Juan) that the differential Galoisgroup of $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $F\langle\boldsymbol{a} \rangle$ is $G(C)$. In this paper we construct a differentialfield extension $\mathcal{L}$ of $F\langle \boldsymbol{a} \rangle$ such thatthe field of constants of $\mathcal{L}$ is $C$, the differential Galois groupof $\boldsymbol{y}'=A(\boldsymbol{a})\boldsymbol{y}$ over $\mathcal{L}$ isstill 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 $A(\boldsymbol{a})$.

Normal Forms in Differential Galois Theory for the Classical Groups