Adjoint orbits in the Lie algebra of the generalized real orthogonal group

Let $(V,γ)$ be a real finite dimensional vector space with a symmetric bilinear form $γ$ whose kernel is spanned by a nonzero vector. The set of invertible real linear mappings of $(V, γ)$ into itself forms a Lie group called the generalized orthogonal group. This paper finds a unique representative for each orbit of the adjoint action of the generalized orthogonal group on its Lie algebra.

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