Irreducible carpets of additive subgroups of type $G_2$ over a field of characteristic $0$

It is proved that any irreducible carpet of type $G_2$ over a field $F$ of characteristic $0$, at least one additive subgroup of which is an $R$-module, where $F$ is an algebraic extension of the field $R$, up to conjugation by a diagonal element defines a Chevalley group of type $G_2$ over an intermediate subfield between $R$ and $F$.