I'm trying to understand the proof of Proposition I.1.9 in the book "Analysis on Symmetric Cones" by Faraut and Koranyi.
The situation is as follows. The connected compact Lie group $G_e$ and the Lie subgroup $K$ have the same associated Lie algebras. The book now claims that $G_e=K$.
What is the simplest way to understand this claim, using the least amount of Lie theory as possible?
The subgroup-subalgebras theorem: if $G$ has Lie algebra $\mathfrak{g}$ and $\mathfrak{h}\leq \mathfrak{g}$ is a Lie subalgebra then there exists a unique connected subgroup $H\leq G$ with Lie algebra $H$. In your case, since $G_e$ is is connected it is the unique subgroup of itself with its Lie algebra.