Bilinear forms on $\mathfrak{sl}_2$-modules and a hypergeometric identity

In this paper we study properties of a certain bilinear form on finite dimensional $\mathfrak{sl}_2(\mathbb{R})$-modules, and how these properties behave with respect to tensor products of modules. An attempt to determine the signature of this form leads to an interesting identity for the hypergeometric series ${}_3F_2$, which is known as the Karlsson-Minton identity.