Witt's theorem

Suppose you have a geometry given by a quadratic form, and you find a rigid match between two of its subspaces — an isometry carrying one onto the other. Witt's theorem promises this partial symmetry is never an accident of the small picture: it can always be enlarged to a rigid symmetry of the whole space. Local congruence of subspaces lifts to global congruence.

Statement (Witt's extension theorem): let (V, Q) be a finite-dimensional nondegenerate quadratic space over a field of characteristic not 2, and let U, U' be subspaces with an isometry σ : U → U' (a linear bijection preserving the form). Then σ extends to an isometry of all of V, i.e. an element of the orthogonal group O(Q) restricting to σ on U. A companion result, Witt's cancellation theorem, says that if Q_1 ⊕ H ≅ Q_2 ⊕ H then Q_1 ≅ Q_2: a common summand may be cancelled.

These theorems are the engine of the structure theory. Cancellation makes the set of forms into a well-behaved monoid and underlies the Witt ring. Extension forces every maximal totally isotropic subspace to have the same dimension (the Witt index) and yields the Witt decomposition of any form into hyperbolic planes plus a unique anisotropic core. The characteristic-2 case is more delicate and was handled separately by Arf and others.