$\DeclareMathOperator\SU{SU}$Let $M$ be a coadjoint orbit of dimension 6 of $\SU(3)$, and let $T$ be the maximal torus in $\SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map associated to the action of $T$ on $M$, then the image of the moment map is a hexagon with vertices $A$, $B$, $C$, $D$, $E$, $F$ the images of the elements $M^T$ by $\mu $.

For $P \subset \mathfrak{t}^*$ an affine space with vectorial direction $\overrightarrow{P}$, let $P^\perp \mathrel{:=} \lbrace \xi \in \mathfrak{t} \mathrel| \langle y, \xi \rangle =0, \forall y \in {(\overrightarrow{P})}^\perp\rbrace $, and let $T_P$ be the sub-torus generated by $\operatorname{Exp}(P^\perp)$.

If $\Sigma \mathrel{:=} \lbrace \text{$P$ convex polytope in $\mathfrak{t}^*$} \mathrel| \exists \text{$Z$ connected component of $M^{T_P}$ s.t $ \mu (Z)= P$}\rbrace$, how can I prove that $\Sigma = \lbrace\text{faces of $ \mu (M)$}\rbrace \cup\lbrace [AD],[BE], [FC]\rbrace$?