The isotropy index of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.
An important example of an isotropic form over the reals occurs in pseudo-Euclidean space.
If we consider the general element (x, y) of V, then the quadratic forms q = xy and r = x2 − y2 are equivalent since there is a linear transformation on V that makes q look like r, and vice versa.
The notation ⟨1⟩ ⊕ ⟨−1⟩ has been used by Milnor and Husemoller[1]: 9 for the hyperbolic plane as the signs of the terms of the bivariate polynomial r are exhibited.
For a general field F, classification of definite quadratic forms is a nontrivial problem.