On irreducible carpets of additive subgroups of type F4

The article describes irreducible carpets A={Ar: r∈Φ} of type F4 over the field K, all of whose additive subgroups Ar are R-modules, where K is an algebraic extension of the field R. An interesting fact is that carpets which are parametrized by a pair of additive subgroups appear only in characteristic 2. Up to conjugation by a diagonal element from the corresponding Chevalley group, this pair of additive subgroups becomes fields, but they may be different. In addition, we establish that such carpets A are closed. Previously, V. M. Levchuk described irreducible Lie type carpets of rank greater than 1 over the field K, at least one of whose additive subgroups is an R-module, where K is an algebraic extension of the field R, under the assumption that the characteristic of the field K is different from 0 and 2 for types Bl, Cl, F4, while for type G2 it is different from 0, 2, and 3 [1]. For these characteristics, up to conjugation by a diagonal element, all additive subgroups of such carpets coincide with one intermediate subfield between R and K.