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For a finite-dimensional Lie algebra representation $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$, its trace form is the symmetric bilinear formIt is invariant: $([x,y],z{)}_V=(x,[y,z]{)}_V$.

The trace form of a number field $K$ is the symmetric bilinear form

The pairing uses the field trace of the product, rather than taking a trace of each factor separately.

For $n\ge 2$, contraction with the invariant symmetric bilinear form splits off the trivial line in the symmetric square of the defining special orthogonal group representation. Its null space is the traceless symmetric square. It contains a highest-weight vector of weight $2e_1$, and the Weyl dimension formula gives $(n+1)(2n-1)$, exactly its dimension. Complete reducibility of compact-group representations therefore makes that null space irreducible. The entire symmetric square is reducible because of the invariant line.

The vector representation is the defining action of a matrix group on the vector space on which its matrices act. For an orthogonal or Lorentz group, it preserves the corresponding symmetric bilinear form.

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]

An isometry of a space with a form between subspaces of a finite-dimensional nonsingular classical space extends to an ambient isometry. This includes alternating bilinear forms in all characteristics, symmetric bilinear forms in characteristic different from two, Hermitian forms with nontrivial involution, and quadratic forms with nonsingular polar form. No nonsingularity of the restricted subspace is required. Bare nonalternating symmetric bilinear forms in characteristic two require additional hypotheses and must not be confused with quadratic spaces.