On the Irreducible Carpets of Additive Subgroups of Type $ F_{4} $

We consider the irreducible carpets \( \mathfrak{A}=\{\mathfrak{A}_{r}:\ r\in\Phi\} \) of type \( F_{4} \) over an algebraical extension \( K \) of a field \( R \) such that all additive subgroups \( \mathfrak{A}_{r} \) are \( R \)-modules. The carpets, parametrized by a pair of additive subgroups, appear only in characteristic 2. This pair of additive subgroups presents (possibly different) fields up to conjugation by a diagonal element in the corresponding Chevalley group. Moreover, we establish that such carpets \( \mathfrak{A} \) are closed. Using Levchuk’s description of the irreducible carpets of Lie type of rank greater than 1 over \( K \), we show that all additive subgroups of the carpets coincide with an intermediate subfield between \( R \) and \( K \) of the carpets of types \( B_{l} \), \( C_{l} \), and \( F_{4} \) in case of the characteristic of \( K \) is not 0 and 2 whereas it is neither 0, 2, nor 3 for type \( G_{2} \) up to conjugation by a diagonal element.