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 $G$ the connected component of the identity in $G(Ω)$ (the automorphisms of the cone $\Omega$). Now consider $K := G \cap O(V)$ where $O(V) := \{T \in \mathrm{GL(V)} \colon T^* = T^{-1}\}$. There is an element $e \in \Omega$ such that $K$ equals the stabilizer subgroup $G_e$ of $G$. For $G$ and $K$ we have the corresponding Lie algebras $\mathfrak{g}$ of $G$ and $\mathfrak{k}$ of $K$, respectively. It turns out that $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ where $\mathfrak{p} := \{T \in \mathfrak{g} \colon T^* = T\}$. Since $\Omega$ is symmetric, $G$ acts transitively on $\Omega$, that is $G \cdot e = \Omega$. Furthermore, as $\mathfrak{k} \cdot e = 0$, we have $\mathfrak{g}\cdot e = \mathfrak{p}\cdot e$.
Could someone please help me understand why this implies $\exp \mathfrak{g} \cdot e = \exp \mathfrak{p} \cdot e$? If possible using the least amount of Lie theory. I'm not an expert in the theory of Lie groups or Lie algebras.
