# Handbook of Pseudo-Riemannian Geometry and Supersymmetry

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#### pp: 629–651

#### DOI: 10.4171/079-1/18

^{[1]}, Thomas Leistner and Thomas Neukirchner

^{[2]}(1) Politecnico di Torino, Italy

(2) Humboldt-Universität zu Berlin, Germany

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` ⊂ GL(`n`,ℝ) 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.

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