Let \(R\) be a commutative ring in which 2 is invertible. A quadratic form on an \(R\)-module \(Q\) is a map \(q:Q\rightarrow R\) such that \(q(ax)=a^2q(x)\) for \(a\in R\) and \(x\in Q\), and \(B_q(x,y)=q(x+y)-q(x)-q(y)\) is a symmetric \(R\)-bilinear map from \(Q\times Q\rightarrow R\). A quadratic \(R\)-module is a pair \((Q,q)\), where \(Q\) is an \(R\)-module, and \(q\) is a quadratic form on \(Q\). We say that a quadratic \(R\)-module \((Q, q)\) is non-singular if the map \(d_{B_q}:Q\rightarrow Q^*\) induced by the bilinear form \(B_q\) is an isomorphism. A quadratic space over \(R\) is a non-degenerate (or non-singular) quadratic \(R\)-module \((Q,q)\), where \(Q\) is a finitely generated projective \(R\)-module. Let \((Q,q)\) be a quadratic \(R\)-space with associated bilinear form \(B_q\) and \(P\) be a finitely generated projective \(R\)-module. The module \(P \oplus P^*\) has a natural quadratic form given by \(p((x, f))=f(x)\) for \(x\in P\), \(f\in P^*\). The corresponding bilinear form \(B_p\) is given by \(B_p((x_1, f_1), (x_2, f_2)) = f_1(x_2) + f_2(x_1)\) for \(x_1, x_2 \in P\) and \(f_1, f_2 \in P^*\). The quadratic space \((P \oplus P^*, p)\), denoted by \(\mathbb{H}(P)\), is called the hyperbolic space of \(P\). Given any homomorphism \(\alpha:Q\rightarrow P\), define \(\alpha^*:P^* \rightarrow Q\) by the formula \(\alpha^*=d^{-1}_{B_q}\circ\alpha^t\), where \(\alpha^t\) denotes the dual map \(P^* \rightarrow Q^*\). If \(\beta:Q\rightarrow P^*\), then define \(\beta^*:P \rightarrow Q\) by precomposing \(d^{-1}_{B_q}\circ \beta^t\) with \(\varepsilon : P \rightarrow P^{**}\), where \(\beta^t\) denotes the dual map \(P^{**} \rightarrow Q^*\). The linear map \(\alpha^*\) is characterized by the relation \((f \circ \alpha)(z) = B_q (\alpha^*(f), z)\) for \(f \in P^*\), \(z\in Q\).
Let \(\operatorname{O}_R(Q)\) denote the orthogonal group of the quadratic module \((Q, q)\). That is, \(\operatorname{O}_R(Q) = \{\theta \in Aut_R(Q) : q(\theta(z)) = q(z) \ \text{for all} \ z \in Q\}\). In [J. Algebra 10, 286–298 (1968; Zbl 0181.04302)], A. Roy defined the elementary transformations \(E_\alpha\) and \(E^*_\beta\) of \(Q \perp \mathbb{H}(P)\) given by \(E_\alpha(z) = z + \alpha(z)\), \(E^*_\beta(z) = z + \beta(z)\), \(E_\alpha(x) =x\), \(E^*_\beta(x) = -\beta^*(x) + x - \frac{1}{2}\beta \beta^*(x)\), \(E_\alpha(f) = -\alpha^*(f) - \frac{1}{2}\alpha \alpha^*(f) +f\), and \(E^*_\beta(f) = f\) for \(z \in Q\), \(x \in P\) and \(f \in P^*\). These transformations are orthogonal transformations. The Dickson-Siegel-Eichler-Roy’s (DSER) subgroup \(\operatorname{EO}_R(Q, \mathbb{H}(P))\) of the orthogonal group \(\operatorname{O}_R(Q\perp \mathbb{H}(P))\) is the subgroup generated by the elementary generators \(E_\alpha\), \(E^*_\beta\), where \(\alpha \in \operatorname{Hom}(Q, P)\) and \(\beta \in \operatorname{Hom}(Q, P^*)\).
The main results of this paper are Theorem 1.1., which states that \(\operatorname{EO}_R(Q, \mathbb{H}(R)^m)\) is normal in \(\operatorname{O}_R(Q\perp \mathbb{H}(R)^m)\), where \(Q\) and \(\mathbb{H}(R)^m\) are quadratic spaces over a commutative ring \(R\) with \(\operatorname{rank}(Q) \geq 1\) and \(m \geq 2\), and Theorem 1.2., which states that \(\operatorname{EO}_R(Q, \mathbb{H}(P))\) is a normal subgroup of \(\operatorname{O}_R(Q\perp \mathbb{H}(P))\), where \(Q\) and \(\mathbb{H}(P)\) are quadratic spaces over a commutative ring \(R\), with \(\operatorname{rank}(Q) \geq 1\) and \(\operatorname{rank}(P) \geq 2\). In order to prove these theorems, the authors prove an analogue of Quillen’s Local-Global principle in the case of extended module for the DSER elementary orthogonal group.