We looking at the definition <the orthogonal group is the group of all matrices that preserve the dot product>, we notice that the <dot product> is one example of <positive definite symmetric bilinear form>, which in turn can also be represented by a matrix as shown at: <matrix representation of a symmetric bilinear form>{full}.

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.:
* $I_2$: another <positive definite symmetric bilinear form> such as $(x_1, x_2)^T(y_1, y_2) = 2 x_1 y_1 + x_2 y_2$?
* $I_-$ what if we drop the <positive definite> requirement, e.g. $(x_1, x_2)^T(y_1, y_2) = - x_1 y_1 + x_2 y_2$?
The answers to those questions are given by the <Sylvester's law of inertia> at <all indefinite orthogonal groups of matrices of equal metric signature are isomorphic>{full}.


Discussion on "What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms" by Ciro Santilli (@cirosantilli, 34)