On pairs of additive subgroups associated with intermediate subgroups of groups of Lie type over nonperfect fields

The author has previously (Trudy IMM UrO RAN, 19(2013), no. 3) described the groups lying between twisted Chevalley groups $G(K)$ and $G(F)$ of type $^2A_l$, $^2D_l$, $^2E_6$, $^3D_4$ in the case when the larger field $F$ is an algebraic extension of the smaller nonperfect field $K$ of exceptional characteristic for the group $G(F)$ (characteristics $2$ and $3$ for the type $^3D_4$ and only $2$ for other types). It turned out that apart from, perhaps, the type $^2D_l$, such intermediate subgroups are standard, that is, they are exhausted by the groups $G(P)H$ for some intermediate subfield $P$, $K\subseteq P\subseteq F$, and of the diagonal subgroup $H$ normalizing the group $G(P)$. In this note, it is established that intermediate subgroups are also standard for the type $^2D_l$.