Groups with $\mathsf{BC}_\ell$-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 $\mathsf{BC}_\ell$ 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 $\mathsf{BC}_\ell$ and satisfying natural conditions, then there is a homomorphism $\mathrm{StU}(R, Δ) \to G$ inducing isomorphisms on the root subgroups, where $\mathrm{StU}(R, Δ)$ is the odd unitary Steinberg group constructed by an odd form ring $(R, Δ)$ with a Peirce decomposition. For groups with root subgroups indexed by $\mathsf A_\ell$ (the already known case) the resulting odd form ring is essentially a generalized matrix ring.