On irreducible carpets of additive subgroups of type $F_4$

The article describes irreducible carpets $\mathfrak{A}=\{\mathfrak{A}_r:\ r\in \Phi\}$ of type $F_4$ over the field $K$, all of whose additive subgroups $\mathfrak{A}_r$ 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 $\mathfrak{A}$ 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 $B_l$, $C_l$, $F_4$, while for type $G_2$ 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 $K$.