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Charles Weibel authored an exposition of bilinear forms over a field. It develops diagonalization and the classification of symmetric bilinear forms, including the real signature described by Sylvester's law of inertia. These classifications explain why changing a basis can simplify a form without removing invariants such as its numbers of positive and negative squares.^[ref]

For a field $K$ of characteristic different from two, the Witt group $W(K)$ classifies nondegenerate symmetric bilinear forms after hyperbolic parts are discarded. Two forms represent the same class if adding suitable hyperbolic planes of a quadratic form makes them isometric. The group operation is orthogonal direct sum.

A metabolic bilinear form has a totally isotropic subspace of half its dimension. In this characteristic it is a sum of hyperbolic planes of a quadratic form and represents zero. The inverse element of the equivalence class of $B$ is represented by $-B$: in $B\oplus (-B)$, the vectors $(v,v)$ form such a half-dimensional totally isotropic subspace.

Over the real numbers, Sylvester's law of inertia reduces a nonsingular form to $p$ positive squares and $q$ negative squares. A positive-negative pair is a hyperbolic plane of a quadratic form, so cancelling pairs leaves exactly the signature $p-q$. Conversely, equal signatures leave the same remainder. Thusunder the signature. This records how much definite form remains after all positive-negative cancellation.^[ref]

Over the complex numbers, every nonzero diagonal coefficient can be rescaled to one, since it has a square root. Two unit squares already form a hyperbolic plane of a quadratic form, using the independent isotropic vectors $(1,i)$ and $(1,-i)$. Pairs therefore cancel, and only the parity of the dimension remains:Odd dimension cannot disappear, because adjoining a hyperbolic plane of a quadratic form changes dimension by two. These are symmetric bilinear forms over the complex numbers, rather than Hermitian forms; confusing the two would give a different classification.^[ref]