Let $K$ be a compact Lie group with Lie algebra $\mathfrak{k}_0$. Suppose that $\sigma$ is an involutive automorphism of $K$ which defines a symmetric pair $(\mathfrak{k}_0,\mathfrak{k}_0^\sigma)$. Take a maximal abelian subspace $\mathfrak{a}_0$ in $\mathfrak{k}_0^{-\sigma}:=\{X\in\mathfrak{k}_0\mid\sigma X=-X\}$, and extend it to a Cartan subalgebra $\mathfrak{t}_0$ of $\mathfrak{k}_0$.
Drop the subscript of each Lie algebra to denote its complexification. Choose compatible positive systems $\Delta^+(\mathfrak{k},\mathfrak{t})$ and $\Delta^+(\mathfrak{k},\mathfrak{a})$ respectively such that the restriction of $\Delta^+(\mathfrak{k},\mathfrak{t})$ to $\mathfrak{a}^*$ is contained in $\Delta^+(\mathfrak{k},\mathfrak{a})$. Denote by $C(\subset\sqrt{-1}\mathfrak{t}_0^*)$ the dominant Weyl chamber with respect to $\Delta^+(\mathfrak{k},\mathfrak{t})$. Put $C_\mathfrak{a}:=C\cap\sqrt{-1}\mathfrak{a}_0^*$.
Show that $C\cap\mathrm{Ad}^*(K)\sqrt{-1}({\mathfrak{k}_0^\sigma})^\bot=C_\mathfrak{a}$. Here $\mathrm{Ad}^*$ is the co-adjoint action and $({\mathfrak{k}_0^\sigma})^\bot$ is orthogonal complement of $\mathfrak{k}_0^\sigma$ in the dual space with respect to some $K$-invariant inner product on $\mathfrak{k}_0$.
It is obvious that $C\cap\mathrm{Ad}^*(K)\sqrt{-1}({\mathfrak{k}_0^\sigma})^\bot\supseteq C_\mathfrak{a}$. However, I do not know how to prove the other direction. I shall be grateful if experts here may give me any hints.