Yoga of Commutators in Roy's Elementary Orthogonal Group

Abstract

In this article, we give explicit proofs of certain commutator relations among the elementary generators of the elementary orthogonal group $EO_A(Q\perp H(P))$, where $A$ is a commutative ring, $Q$ is a non-singular quadratic $A$-space and $H(P)$ is the hyperbolic space of a finitely generated projective module $P$ with the natural quadratic form. Using these relations, we established a local-global principle of D. Quillen for the Dickson--Siegel--Eichler--Roy (DSER) elementary orthogonal transformations in \cite{aarr}. In \cite{aa1}, by using these commutator relations, we prove the normality of this elementary group in the orthogonal group under some conditions on the hyperbolic rank. Also, these relations are used to obtain further information about this orthogonal group and in comparing it with similar groups such as Hermitian groups and odd unitary groups.