Another presentation of orthogonal Steinberg groups

Abstract

We use the pro-group approach to show that $\mathrm{StO}(M, q)$ admits van der Kallen's "another presentation", where $M$ is a module over a commutative ring with sufficiently isotropic quadratic form $q$. Moreover, we construct an analog of ESD-transvections in orthogonal Steinberg pro-groups under some assumptions on their parameters.