Quaternary quadratic form

A quadratic form in four variables. A quaternary quadratic form over a field is related to the algebra of quaternions (cf. Quaternion) over the same field. Namely, corresponding to the algebra with basis , , , and , is the quaternary quadratic form which is the norm of the quaternion,

For quaternary quadratic forms corresponding to quaternion algebras, and only for these, composition of quaternary quadratic forms is defined:

where the coordinates of the vector are bilinear forms in and . Composition of this kind is possible only for quadratic forms in two, four and eight variables.

Comments

The last-mentioned result is known as Hurwitz's theorem; see Quadratic form.