What are the classes in $SU(2)/T$ fixed by multiplication by any element in $T$

Question

Let $G = SU(2)$ and let $T $ be the subset of diagonal matrices in $G$.

I'm stuck trying to understand what the following set is:

$$ M= \lbrace [g] \in G/T\mid [tg]=[g], \forall t \in T\rbrace. $$

I just can think of the classes $[t]$, $t \in T$ .

Are there any other classes ?

Answer 1

Hint: The subset of diagonal matrices in $G=SU(2)$ forms a subgroup of $G$. If $$ G/T=\{Tg\,\mid\,g\in G\}, $$ then $M=G/T$. If $$ G/T=\{gT\,\mid\,g\in G\}, $$ then $M=\{[e],[a]\}$ with $$ a= \left(% \begin{array}{rc} 0 & 1 \\ -1 & 0 \\ \end{array}% \right). $$