# Symplectic form of a coadjoint orbit of a compact connected Lie group

Let $$G$$ be a compact Lie group with Lie algebra $$\mathfrak{g}.$$ Denote by $$\mathfrak{g}^*$$ the dual space of $$\mathfrak{g}$$. Let $$r$$ be an element of $$\mathfrak{g}^*$$ such that $$G_r$$ the stabilizer of $$r$$ under the coadjoint action is a maximal torus of $$G$$. Denote by $$\mathcal{O}_r$$ the coadjoint orbit of $$G$$ which pass through $$r$$.

$$\mathcal{O}_r$$ is endowed with a 2-form which is a symplectic form $$\omega_\alpha(\hat{X},\hat{Y})= -\alpha([X,Y]), \alpha \in \mathfrak{g}^*, \quad X,Y, \in \mathfrak{g}.$$ Often it is more convenient to choose an inner product $$\langle . , . \rangle$$ on $$\mathfrak{g}$$ to identify $$\mathfrak{g}^*$$ with $$\mathfrak{g}$$. Once such an inner product has been chosen, we can write the 2-form as $$\omega_\lambda(\hat{X},\hat{Y}) = −\langle \lambda, [X,Y] \rangle, \lambda, X, Y \in \mathfrak{g}.$$

If $$G$$ is semisimple, then we choose the killing form denoted $$k$$ to define $$\omega$$: $$\omega_\lambda(\hat{X},\hat{Y}) = −k(\lambda, [X,Y]).$$

$$\textbf{Question}$$: In the case where $$G$$ is a compact connected Lie group (not necessarily semisimple), why does the 2-form $$\omega_\lambda(\hat{X},\hat{Y}) = −k(\lambda, [X,Y]), \lambda, X, Y \in \mathfrak{g}.$$ defined using the killing form defines a symplectic form on the coadjoint orbit $$\mathcal{O_r}$$ of $$G$$ ?

This a follow-up question of my questions here https://mathoverflow.net/questions/430926/question-about-the-k%c3%a4hler-structure-on-generic-coadjoint-orbits/431200#431200 and https://mathoverflow.net/questions/431284/compact-coadjoint-orbits?noredirect=1#comment1110075_431284. LSpice explained that If $$G$$ is a compact connected Lie group we may assume that it is semisimple, since there are a compact, semisimple Lie subgroup $$G'$$ of $$G$$, the derived group of $$G$$, and a torus $$T$$ in $$G$$, the maximal central torus in $$G$$, such that the multiplication map $$G' \times T \to G$$ is a covering map. Could anyone give more details about this please?