This paper supplements a previous paper by the author (Overgroups of subsystem subgroups in exceptional groups: a 2A1-proof, (2020)), where the over-group lattice of the elementary subsystem subgroup E(& UDelta;, R) of the Chevalley group G(& phi;, R) for a sufficiently large root subsystem & UDelta; was studied. Currently, the subject-matter is the relationship between the elementary subgroup Ep(& sigma;) given by a net of ideals & sigma; of the ring R, and the stabilizer S(& sigma;) of the corresponding Lie subalgebra of the Chevalley algebra. In particular, it is proved that under a certain condition the subgroup Ep(& sigma;) is normal in S(& sigma;), and some properties of the corresponding quotient group are established.