Non-transitive subgroups of co-rank one in the orthogonal group

It is known that any non-transitive closed subgroup $G$ in the orthogonal group is remetri\-zable by a non-Euclidean Minkowski functional keeping the elements in $G$ as linear isometries. Minkowski geometry is an alternative of Euclidean geometry for $G$. To measure the non-transitivity of the subgroup in the orthogonal group, we can use the Hausdorff distance between the Euclidean unit sphere and the orbit of a Euclidean unit element under $G$. The so-called flat subspace is spanned by the confi\-guration of elements where the Hausdorff distance is attained at. Taking the maximum of the dimension of flat subspaces, we have the rank of the group $G$. It is known that subgroups of maximal rank are finite or reducible. They are natural prototypes of non-transitive subgroups in the orthogonal group. In the paper, we are going to investigate non-transitive subgroups of rank $n-1$ (co-rank one). We prove that the unit component $G^0 \subset G$ must be an Abelian subgroup in the special orthogonal group, and the elements in $G$ can be described in terms of subspaces given by simultaneous quasi-diagonalization in $G^0$. Independently of the rank condition, applications of simultaneous quasi-diagonalization are also presented for one- and two-dimensional non-transitive closed subgroups.