Groups with ${\mathsf{B}\mathsf{C}}_{\mathrm{\ell }}$-commutator relations

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