Subgroups of split orthogonal groups over a commutative ring

We describe subgroups of the split orthogonal group $\Gamma=\mathrm{GO}(n,R)$ of degree $n$ over a commutative ring $R$ such that $2\in R^*$, which contain the elementary subgroup of a regularly embedded semi-simple group $F$. We show that if the ranks of all irreducible components of $F$ are at least 4, then the description of its overgroups is standard in the sense that for any intermediate subgroup $H$ there exists a unique orthogonal net of ideals such that $H$ lies between the corresponding net subgroup and its normalaser in $\Gamma$. A similar result for subgroups of the general linear group $\mathrm{GL}(n,R)$ with irreducible components of ranks at least 2 was obtained by Z. I. Borevich and the present author. We construct examples which show that if $F$ has irreducible components of ranks 2 or 3, then the standard description does not hold. The paper is based on the previous publications by the author, where similar results were obtained in some special cases, but the proof is based on a new computational trick.