Given a matrix $A$ with metric signature containing $m$ positive and $n$ negative entries, the indefinite orthogonal group is the set of all matrices that preserve the associated bilinear form, i.e.:
Note that if $A=I$, we just have the standard dot product, and that subcase corresponds to the following definition of the orthogonal group: Section "The orthogonal group is the group of all matrices that preserve the dot product".

$O(m,n)=O∈M(m+n)∣∀x,yx_{T}Ay=(Ox)_{T}A(Oy)$

As shown at all indefinite orthogonal groups of matrices of equal metric signature are isomorphic, due to the Sylvester's law of inertia, only the metric signature of $A$ matters. E.g., if we take two different matrices with the same metric signature such as:
and:
both produce isomorphic spaces. So it is customary to just always pick the matrix with only +1 and -1 as entries.

$[100−1 ]$

$[200−3 ]$