Let $G$ be a semisimple Lie group and let $\frak p\oplus t$ be a Cartan decomposition of $\frak g$ and $K$ the connected subgroup with Lie algebra $\frak t$. Choose a maximal abelian subalgebra $\frak a \subset p$, an ordering of $\frak a^\ast$, and let $\Sigma^+$ be the positive roots of $\frak a$ with respect to $\frak g$. Denote by $\frak a^+$ the positive Weyl chamber, $$\mathfrak{a}^+=\{ H \in \mathfrak{a} : \lambda (H)>0 \hspace{3mm} \lambda \in \Sigma^+ \}$$

I read that every $Ad(K)$ orbit in $\frak p$ intersects $\frak \overline{a^+}$ exactly once. Why is this?

Edit: I'm aware that $$\bigcup_{k\in K} Ad(k)\frak a= p$$ so all I need to know is why any $H\in \frak a$ has a unique conjugate in $\overline{\frak a^+}$.


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