Orthogonal linear group-subgroup pairs with the same invariants

The main theorem of Galois theory implies that there are no finite group-subgroup pairs with the same invariants. On the other hand, if we consider linear reductive groups instead of finite groups, the analogous statement is no longer true: There exist counterexample group-subgroup pairs with the same invariants. However, it is possible to classify all these counterexamples for certain types of groups. In [S. Solomon, Irreducible linear group-subgroup pairs with the same invariants, J. Lie Theory 15 (2005), 105-123], we provided the classification for connected complex irreducible groups, and, in this paper, for connected complex reductive orthogonal groups, i.e., groups that preserve some nondegenerate quadratic form.