7.28 (1980)

Let $G(K)$ be the Chevalley group over a commutative ring $K$ associated with the root system $\Phi$ as defined in (R. Steinberg, Lectures on Chevalley groups, Yale Univ., New Haven, Conn., 1967). This group is generated by the root subgroups $x_r(K)$, $r \in \Phi$. We define an elementary carpet of type $\Phi$ over $K$ to be any collection of additive subgroups $\{\mathfrak{A}_r \mid r \in \Phi\}$ of $K$ satisfying the condition
$$c_{ij,rs}\mathfrak{A}_r^i\mathfrak{A}_s^j \subseteq \mathfrak{A}_{ir+js} \quad \text{for } r, s, ir + js \in \Phi, \ i > 0, \ j > 0,$$ where $c_{ij,rs}$ are constants defined by the Chevalley commutator formula and $\mathfrak{A}_r^i = \{a^i \mid a \in \mathfrak{A}_r\}$. What are necessary and sufficient conditions (in terms of the $\mathfrak{A}_r$) on the elementary carpet to ensure that the subgroup $\langle x_r(\mathfrak{A}_r) \mid r \in \Phi \rangle$ of $G(K)$ intersects with $x_r(K)$ in $x_r(\mathfrak{A}_r)$? See also 15.46.