Tensor product of quadratic forms

In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as "quadratic spaces". If "R" is a commutative ring where 2 is invertible (that is, "R" has characteristic formula_1), and if formula_2 and formula_3 are two quadratic spaces over "R", then their tensor product formula_4 is the quadratic space whose underlying "R"-module is the tensor product formula_5 of "R"-modules and whose quadratic form is the quadratic form associated to the tensor product of the bilinear forms associated to formula_6 and formula_7.
In particular, the form formula_8 satisfies
(which does uniquely characterize it however). It follows from this that if the quadratic forms are diagonalizable (which is always possible if 2 is invertible in "R"), i.e., 
then the tensor product has diagonalization