  Let $(V,\gamma )$ be a real finite dimensional vector space with a symmetric
bilinear form $\gamma $ whose kernel is spanned by a nonzero vector. The set of
invertible real linear mappings of $(V, \gamma )$ 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.
