Let (V, b) be a finite-dimensional vector space over a field k of characteristic different from 2 together with a non-degenerate symmetric or skew-symmetric bilinear form. If f : U → U′ is an isometry between two subspaces of V then f extends to an isometry of V.
Witt's theorem implies that the dimension of a maximal totally isotropic subspace (null space) of V is an invariant, called the index or Witt index of b, and moreover, that the isometry group of (V, b) acts transitively on the set of maximal isotropic subspaces. This fact plays an important role in the structure theory and representation theory of the isometry group and in the theory of reductive dual pairs.
Let (V, q), (V1, q1), (V2, q2) be three quadratic spaces over a field k. Assume that
Then the quadratic spaces (V1, q1) and (V2, q2) are isometric:
In other words, the direct summand (V, q) appearing in both sides of an isomorphism between quadratic spaces may be "cancelled".