Yoga of commutators in DSER elementary orthogonal group

Abstract

In this article, we consider the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal subgroup of the orthogonal group of a non-degenerate quadratic space with a hyperbolic summand over a commutative ring, introduced by Roy. We prove a set of commutator relations among the elementary generators of the DSER elementary orthogonal group. As an application, we prove that this group is perfect and an action version of the Quillen’s local-global principle for this group is proved. This affirmatively answers a question of Rao in his Ph.D. thesis.