Given a quadratic form $q$ on $\mathbb R^n$, are there methods to determine the number of "standardized" quadratic forms using the Gauss's square decomposition algorithm (https://fr.wikipedia.org/wiki/Réduction_de_Gauss) ?
In general, using this algorithm can lead to several standardized forms. I was wondering if there was a sort of "canonical" representative for the quadratic form which after a "change of basis" gives all the other ones ?
For instance, on $\mathbb R^4$ for the quadratic form $q(x,y,z,t) = xt - 2yz - yt + xz$, we can quickly give two standardized forms (after using Gauss's square decomposition algorithm).
We obtain the following four independent linear forms :
$f_1 = x - y + z + t$ ; $f_2 = x - y - z - t$ ; $f_3 = y + z$ ; $f_4 = y - z$.
And also :
$f_1 = x + z + t$ ; $f_2 = x - z - t$ ; $f_3 = y + 2z + t$ ; $f_4 = y - 2z - t$.
Are there references about these kind of questions ?
Thanks in advance !