The genetics of universal Chevalley groups over some commutative rings (Q1916575)

Let \(\mathcal L\) be a simple complex-valued \(\Phi\)-type Lie algebra, \(R\) a commutative ring with identity, \(G(\Phi, R)\) the universal Chevalley group over \(R\) corresponding to \(\mathcal L\), and \(E(\Phi,R)\) the elementary subgroup of \(G(\Phi,R)\) generated by all \(x_\alpha(t)\), \(\alpha\in\Phi\), \(t\in R\). For invertible \(t\), let \(w_\alpha(t)=x_\alpha(t)x_{-\alpha}(-t^{-1})x_\alpha(t)\) and \(h_\alpha(t)=w_\alpha(t)w_\alpha(-1)\). It is well known that the following relations hold: \[ x^{-1}_\alpha(t)x^{-1}_\alpha(u)x_\alpha(t+u)=1,\tag{A} \] \[ x_\beta(t)x_\alpha(u)x^{-1}_\beta(t)x^{-1}_\alpha(u)\prod_{i,j>0}x_{i\alpha+j\beta}(c_{ij}t^iu^j)=1,\tag{B} \] \[ w_\alpha(t)x_\alpha(u)w^{-1}_\alpha(t)x^{-1}_{-\alpha}(-t^{-2} u)=1,\tag{B\('\)} \] \[ h_\alpha(t)h_\alpha(u)h^{-1}_\alpha(tu)=1,\quad\text{ and }\quad\text{(D)}\quad x_\alpha(t)x_{-\alpha}(u)[x_{-\alpha}(u/p)h_\alpha(p)x_\alpha(t/p)]^{-1}=1,\tag{C} \] where \(p=1+tu\) with \(tu\in\text{rad }R\). The author proves that if \(R/\text{rad }R=\prod R/J\) where \(J\) runs over the maximal ideals of \(R\), then \(E(\Phi,R)\) is the group with generators \(x_\alpha(t)\), \(\alpha\in\Phi\), \(t\in R\), subject to the relations (A), \((\text{B}')\), (C) and (D) for the case \(\text{rank }\Phi=1\), and subject to the relations (A), (B), (C) and (D) for the case \(\text{rank }\Phi\geq 2\). This extends the corresponding result over semilocal rings.