Overgroups of subsystem subgroups in exceptional groups: levels

An embedding of root systems $\Delta\subseteq\Phi$ determines the corresponding regular embedding $G(\Delta,R)\le G(\Phi,R)$ of Chevalley groups, over an arbitrary commutative ring $R$. Denote by $E(\Delta,R)$ the elementary subgroup of $G(\Delta,R)$. In the present paper we initiate the study of intermediate subgroups $H$, $E(\Delta,R)\le H\le G(\Phi,R)$, provided that $\Phi=\mathrm{E_6,E_7,E_8,F}_4$ or $\mathrm G_2$, and there are no roots in $\Phi$ orthogonal to all of $\Delta$. There are 72 such pairs $(\Phi,\Delta)$. For $\mathrm F_4$ and $\mathrm G_2$ we assume, moreover, that $2\in R^*$ or $6\in R^*$, respectively. For all such subsystems $\Delta$ we construct the levels of intermediate subgroups. We prove that these levels are detemined by certain systems of ideals in $R$, one for each $\Delta$-equivalence class of roots in $\Phi\setminus\Delta$, and calculate all relations among these ideals, in each case.