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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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Question about action of exponential of Lie algebras (Faraut and Koranyi's book)

I'm trying to understand a statement on page 49 in the book "Analysis on Symmetric Cones" by Faraut and Koranyi. The situation is as follows. Let $\Omega$ be a symmetric cone in $V$ and denote by $G$ …