On conjugacy in a Chevalley group of large Abelian subgroups of the unipotent subgroup

Let $U$ be the unipotent subgroup of a Chevalley group over a finite field. The well-known problem about describing the set of “large” (of maximal order) Abelian subgroups in $U$ of exceptional type is investigated. The description of normal large Abelian subgroups in $U$ was established earlier. It is proved that each large Abelian subgroup from $U$ is conjugate in the Chevalley group of type $F_4$ over a finite field of characteristic not equal to 2 to a normal subgroup in $U$. It is shown that for the groups $U$ of type $G_2$ and $^3D_4$ the similar conclusion is not true.