Let $M$ be a symmetric space of the noncompact or compact type with $G := \text{Isom}(M)^0$. It's Weyl group is defined as $$ W := N_G(A) / Z_G(A), $$ where $A \leq G$ is a maximal abelian subgroup. One can show that $W \cong N_K(A)/Z_K(A)$ where $K = G_p$ is the stabilizer of a point $p \in M$.
The intuition on the Weyl group is that if $F \subseteq M$ is a maximal flat, then the isometries in $\text{Isom}(F) \cong \mathbb{R}^r \rtimes O(r)$ which are coming from isometries of $G$ are exactly those whose orthogonal part is in $W$. This means that the image of the map $$ G_F :=\{ g \in G : gF =F\} \to \text{Isom}(F). $$ should be isomorphic to $A \rtimes W$.
Questions:
- How can I construct such an isomorphism explicitly?
- Is $G_F$ itself a semidirect product of something with $W$?
I have tried several variations, for example constructing a split short exact sequence with the image of the above map in the middle, but It doesn't work out. I also couldnt find a treatment of this relationship in the books by Helgason or Ebelrein where this isomorphism is constructed.