  The main theorem of Galois theory states that there are no finite
group-subgroup pairs with the same invariants. On the other hand, if we
consider complex 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's possible to
classify all these counterexamples for certain types of groups. In [16], we
provided the classification for connected complex irreducible groups, and, in
this paper, for connected complex orthogonal groups, i.e., groups that preserve
some non-degenerate quadratic form.
