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?