Subgroups of the general linear group containing the elementary subgroup over a commutative ring extension of rank 2

Let $R=\prod_{i\in I}F_i$ be a direct product of fields and let $S=R[\sqrt d]=\prod_{i\in I}F_i[\sqrt{d_i}]$ be a ring extension of rank 2 of $R$. The subgroups of the general linear group $\operatorname{GL}(2n,R)$, $n\geq3$ that contain the elementary group $E(n,S)$ are described. It is shown that for every such a subgroup $H$ there exists a unique ideal $A\unlhd R$ such that
$$ E(n,S)E(2n,R,A)\leq H\leq N_{\operatorname{GL}(2n,R)}(E(n,S)E(2n,R,A)). $$