Some non-maximal arithmetic groups

Let k be a non-finite Dedekind domain, and σ be the ring of its integers. We shall assume that the ring R = σ/ (2) is finite. Let us denote by M_n (k) (resp. M_n(σ) ) the ring of all n by n matrices with entries in k (resp. in σ), and G^l_n (k) its group of units.

We denote by s^l_n (k) the subgroup of G^l_n (k) whose elements g  have determinant, det g, equal to one. Let  H ε M_n  (σ) be a symmetric matrix, i.e., H = ^tH where ^tH denotes the transpose matrix of H. We let G = SO (H) = { g ε S^l_n (k) l ^tgHg = H }, and we let G_σ = G∩M_n (σ). We want to exhibit certain H for which G_σ is not maxinal in G, in the sense that there exist a subgroup Δ contains G_σ properly and [Δ : G_σ] is finite.