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Automorphism group of indefinite orthogonal Lie group $G=O(p,q)$ vs that of a double covering group $\tilde{G}$

Previously I mentioned in Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? that the automorphism group of a Lie group 𝐺 may be the same as that of ...
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