Normality and $K_1$-stability of Roy's elementary orthogonal group

Abstract

In this paper, we prove the normality of the Roy's elementary orthogonal group (Dickson--Siegel--Eichler--Roy or DSER group) over a commutative ring which was introduced by A. Roy in [MR0231844] under some conditions on the hyperbolic rank. We also establish a stability theorem for $K_1$ of Roy's group. We obtain a decomposition theorem for the elementary orthogonal group which is used to deduce the stability theorem.