15.46 (2002)

Can the question 7.28 on conditions for admissibility of an elementary carpet $\mathfrak{A} = \{\mathfrak{A}_r \mid r \in \Phi\}$ be reduced to Lie rank 1 if $K$ is a field? A carpet $\mathfrak{A}$ of type $\Phi$ of additive subgroups of $K$ is called admissible if in the Chevalley group over $K$ associated with the root system $\Phi$ the subgroup $\langle x_r(\mathfrak{A}_r) \mid r \in \Phi \rangle$ intersects $x_r(K)$ in $x_r(\mathfrak{A}_r)$. More precisely, is it true that the carpet $\mathfrak{A}$ is admissible if and only if the subcarpets $\{\mathfrak{A}_r, \mathfrak{A}_{-r}\}$, $r \in \Phi$, of rank 1 are admissible?