Irreducible carpets of Lie type $B_l$, $C_l$ and $F_4$ over fields

V.M. Levchuk described irreducible carpets of Lie type of rank greater than $1$ over the field $F$, at least one additive subgroup of which is an $R$-module, where $F$ is an algebraic extension of the field $R$, in assumption that the characteristic of the field $F$ is different from $0$ and $2$ for the types $B_l$, $C_l$, $F_4$, and for the type $G_2$ it is different from $0, 2$ and $3$ (Algebra i Logika, 1983, 22, no. 5). It turned out that, up to conjugation by a diagonal element, all additive subgroups of such carpets coincide with one intermediate subfield between $R$ and $F$. We solve a similar problem for carpets of types $B_l$, $C_l$, $F_4$ over a field of characteristic $0$ and $2$. It turned out that carpets appear in characteristic $2$, which are parameterized by a pair of additive subgroups, and for types $B_l$ and $C_l$ one of these two additive subgroups may not be a field.