A fundamental property of Suslin matrices

Abstract

We describe a homomorphism from the group SUm_r (R), generated by Suslin matrices, when r is even, to the special orthogonal group SO_2(r+1) (R) by relating the Suslin matrix corresponding to a pair of vectors v, w, with 〈v, w〉 = 1, to the product of two reflections, one w.r.t. the vectors v, w and the other w.r.t. the vectors e₁, e₁ (of length one). When r is odd we can still associate a product of reflections with an element of SUm_r (R), which is well defined up to a unit u, with u² = 1. This association enables one to study the orbit space of unimodular vectors under the elementary subgroup.