Bireflectionality of the orthogonal group over a valuation domain

Abstract

Let V be a free module of rank n over a valuation domain R. Assume V is endowed with a quadratic form Q such that Q( v) is a unit in R for every unimodular v in V. Let O( V) be the group of all form preserving linear transformations on V. Then every element in O( V) is a product of at most two involutions.