With the information contained in the invariant bilinear forms one can easily list all simple -modules:

It makes it possible to write differential equations on sections of a fiber bundle in an invariant form.

They have a symmetric invariant bilinear form (,).

The main tensorial invariant of a connection form is its curvature form.

It is 0 exactly when the irreducible representation has no invariant bilinear form, which is equivalent to saying that its character is not always real.

It is 1 exactly when the irreducible representation has a symmetric invariant bilinear form.

Thus the free energy density can be expressed as the invariant form of , , and :

The laws of physics can be expressed in a generally invariant form.

Hence it is possible to define the pullbacks of the invariant forms from F(n):

To complete the description of real representation, we must describe how these structures interact with the invariant bilinear forms.