Groups with BCℓ-commutator relations

Abstract

Isotropic odd unitary groups generalize Chevalley groups of classical types over commutative rings and their twisted forms. Such groups have root subgroups parameterized by a root system ${\mathrm{BC}}_{\ell }$ and may be constructed by so-called odd form rings with Peirce decompositions. We show the converse: if a group G has root subgroups indexed by roots of ${\mathrm{BC}}_{\ell }$ and satisfying natural conditions, then there is a homomorphism

inducing isomorphisms on the root subgroups, where

is the odd unitary Steinberg group constructed by an odd form ring $(R,\Delta )$ with a Peirce decomposition. For groups with root subgroups indexed by $A_{\ell }$ (the already known case) the resulting odd form ring is essentially a generalized matrix ring.