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Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
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on the center of a Lie group
You may find complete answer to your question in Wallach, Harmonic Analysis on homogeneous spaces, chapter 4, best regards jorge
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Maximal subgroups of indefinite special orthogonal group SO(p,q)
A Russian mathematician whose name is Komrakov, has listed the maximal subgroups of SO(p,q), if you do not find the articles, i have the pdf fil's, best regards jorge