Invariant bilinear forms of algebras given by faithfully flat descent

The existence of nondegenerate invariant bilinear forms is one of the most important tools in the study of Kac-Moody Lie algebras and extended affine Lie algebras. In practice, these forms are created, or shown to exist, either by assumption or in an ad hoc basis. The purpose of this work is to describe the nature of the space of invariant bilinear forms of certain algebras given by faithfully flat descent (which includes the affine Kac-Moody Lie algebras, as well as Azumaya algebras and multiloop algebras) within a functorial framework. This will allow us to conclude the existence, uniqueness and nature of invariant bilinear forms for many important classes of algebras.