On weakly supplemented carpets of Lie type over commutative rings

Relationships between two hypothetical conditions for the closedness of carpets of Lie type over commutative rings are considered. The results of A. K. Gutnova and V. A. Koibaev (Vestnik of Saint Petersburg University, Mathematics. Mechanics. Astronomy, 2020) on the separation of classes of weakly supplemented and supplemented matrix carpets over fields of characteristic 0 and 2 are carried over to carpets of any Lie type over commutative rings even characteristic. It is established that these classes of carpets are also separated by examples of irreducible closed carpets of type Bl and Cl over nonperfect fields of characteristic 2, parametrized by two additive subgroups, which were constructed in the work of Ya. N. Nuzhin and A. V. Stepanov (Algebra and Analysis, 2019) to obtain non-standard groups between Chevalley groups over a field and its subfield.