Overgroups of subsystem subgroups in exceptional groups: inside a sandwich

This paper is a supplement to the author's paper (Overgroups of subsystem subgroups inin exceptional groups: an $2{A}_1$-proof, (2020)), which was devoted to the study of the lattice of overgroups for the elementary subsyten subgroup $E(\Delta,R)$ in the Chevalley group $G(\Phi,R)$ for a syfficiently large subsystem $\Delta$. The relationship will be studied between the elementary subgroup $\hat{E}(\sigma)$ determined by a net of ideals $\sigma$ of the ring $R$, and the stabilizer $S(\sigma)$ of the corresponding subalgebra in the Chevalley algebra. In particular, it will be proved that under certain conditions the subgroup $\hat{E}(\sigma)$ is normal in $S(\sigma)$, and some properties of the corresponding factor-group will be explored.