This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either there 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.
