Equality of DSER elementary orthogonal group and Eichler-Siegel-Dickson transvection group

TL;DR

The article establishes that over a commutative ring where $2$ is invertible, the DSER elementary orthogonal group ${\rm EO}_{R}(Q,\mathbb{H}(P))$ and the Eichler-Siegel-Dickson transvection group ${\rm TransO}(M,\langle\cdot,\cdot\rangle)$ coincide. The authors first resolve the free-case by treating even and odd ranks separately and then extend the result to the general case using a local-global principle, including relative versions with respect to an ideal $I$. They show that, in the free setting, DSER transformations coincide with elementary transvections, and that relative groups mirror this equality. By leveraging splitting lemmas, commutator identities, and localization techniques (Suslin–Kopeiko-type results), the paper unifies two classical transformation frameworks for orthogonal groups over rings, thereby generalizing prior stability- and equality-type results. This equivalence has implications for the structure and computation of orthogonal groups over rings and for the broader study of stability phenomena in quadratic modules.

Abstract

We prove the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal group, which was introduced by Amit Roy in 1968 and the Eichler-Siegel-Dickson transvection group, which is in literature in the works of Dickson, Siegel and Eichler, are equal over a commutative ring in which $2$ is invertible. We prove the equality in the free case by considering the odd and even case separately and then generalize this result by using the local-global principle. This result generalizes previous results concerning the equality of elementary orthogonal transvection groups.

Theorems & Definitions (55)

• Definition 2.1: Hyperbolic space
• Definition 3.1
• Definition 3.2
• Definition 3.3: Elementary orthogonal transvections
• Lemma 4.1
• proof
• Lemma 4.2: Suresh1994
• Lemma 4.3
• proof
• Definition 5.1