Geometry applications of irreducible representations of Lie Groups

In this note we give proofs of the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup $G \subset Gl(n,\rr)$ is closed. Moreover, if $G$ admits an invariant bilinear form of Lorentzian signature, $G$ is maximal, i.e. it is conjugated to $SO(1,n-1)_0$. We calculate the vector space of $G$-invariant symmetric bilinear forms, show that it is at most $3$-dimensional, and determine the maximal stabilizers for each dimension. Finally, we give some applications and present some open problems.