I've never posted here and never learned much differential geometry, so sorry in advance if I'm doing something wrong.
Let $G$ be a compact Lie-group and $A$ its Cartan-subgroup. I have to show that the manifold $G/A$ is symplectic with the symplectic form $\omega(g,f)= \mathrm{tr}[g,f]\lambda$ with $[\cdot,\cdot]$ being the Lie-bracket, $f,g \in TG$ and a suitable $\lambda \in TA$.
Now my confusion already starts with the definition of ω. Isn't the trace of a commutator always zero?
I think I have to show that $d\omega = 0$ and that $\omega$ is non-degenerate, correct? Do I also have to show that $G/A$ is smooth?
$$d\omega(X,g,f)=X\mathrm{tr}[g,f]\lambda − g\mathrm{tr}[X,f]\lambda + f\mathrm{tr}[X,g]\lambda − \mathrm{tr}[[X,g],f]\lambda + \mathrm{tr}[[X,f],g]\lambda − \mathrm{tr}[[g,f],X]\lambda$$
I think I can use the Jacobi-identity for the last three terms after using that the trace is linear. But I don't know about the other terms. I also have no idea what is meant by "a suitable $\lambda$".
It would be great if someone could help me.
Kind regards