bilinearity

On non-homogeneous elements, the bracket is extended by bilinearity.

The verification crucially uses the bilinearity of the Weil-pairing.

The exterior product of a k-vector with a p-vector is a (k+p)-vector, once again invoking bilinearity.

By the bilinearity of the pairings, the two expressions are equal if and only if modulo the order of .

In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when considering real or complex Lie algebras.

The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.

Together with the skew-symmetry and bilinearity of the cross product, these three identities are sufficient to determine the cross product of any two vectors.

Note that the bilinearity and alternating properties imply anticommutativity, i.e., for all elements x, y in , while anticommutativity only implies the alternating property if the field's characteristic is not 2.

Once we have established a multiplication table, it is then applied to general vectors x and y by expressing x and y in terms of the basis and expanding x x y through bilinearity.

The requirement of sesquilinearity (i.e. linearity in the first argument and antilinearity in the second argument) instead of bilinearity secures that the inner product of a vector with itself is real (due to the conjugate symmetry):