This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either [there](https://math.stackexchange.com/questions/958531/why-do-adk-orbits-in-the-1-eigenspace-of-a-cartan-decomposition-intersect-t) or here.


$G$ is a semisimple Lie group with a choice of Cartan decomposition on its Lie algebra $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$. 
The Cartan involution also induces an inner product on $\mathfrak{g}$ and we denote the sphere in $\mathfrak{p}$ by $\mathfrak{p}\_1$.
Let $K$ be the connected subgroup associated to $\mathfrak{k}$ and let $\mathfrak{a}\subset \mathfrak{p}$ be a maximal abelian subalgebra.


The action of $\mathfrak{a}$ on $\mathfrak{g}$ gives rise to a root system $\Sigma$ on $\mathfrak{a}^\*$.
Fix a notion of positivity here and let 
$$\mathfrak{a}^+ := \lbrace H \in \mathfrak{a}: \lambda(H)>0 \text{ for all } \lambda \in \Sigma^+\rbrace$$


**Question:** Why is it that the unit vectors in ${\mathfrak{a}^+}$ is a transversal for the adjoint action of $K$ on $\mathfrak{p}\_1$?
More precisely, for every $X \in \mathfrak{p}\_1$, there is a unique unit vector $H$ in $\overline{\mathfrak{a}^+}$ which is in the $K$-orbit of $X$. 


$\mathfrak{p} = \bigcup \operatorname{Ad}(k)\mathfrak{a}$, coincidence of the analytic Weyl group and algebraic Weyl group, and the fact that $\overline{\mathfrak{a}^+}$ is a fundamental domain for the algebraic Weyl group acting on $\mathfrak{a}$, together imply that each $K$ orbit in $\mathfrak{p}$ passes through at least once. But it's not yet clear to me why two unit vectors in $\overline{\mathfrak{a}^+}$ cannot be related by some $k \in K\smallsetminus N\_K(\mathfrak{a})$.


Thank you for reading.



A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$
This has been answered [here](https://math.stackexchange.com/questions/958531/why-do-adk-orbits-in-the-1-eigenspace-of-a-cartan-decomposition-intersect-t/3678116#3678116) on MSE.


FYI, this is Theorem 4.21 (without proof) on page 74 in [Bekka and Mayer - Ergodic theory and topological dynamics of group actions on homogeneous spaces](https://www.cambridge.org/core/books/ergodic-theory-and-topological-dynamics-of-group-actions-on-homogeneous-spaces/D44A57C1D660059C5C4AA7FA8E244659).


