What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms

ID: what-happens-to-the-definition-of-the-orthogonal-group-if-we-choose-other-types-of-symmetric-bilinear-forms

By looking at this more general point of view, we could ask ourselves what happens to the group if instead of the dot product we took a more general bilinear form, e.g.:
The answers to those questions are given by the Sylvester's law of inertia at Section "All indefinite orthogonal groups of matrices of equal metric signature are isomorphic".

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