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 thecoadjoint orbit $ \mathcal{O_r}$ of $G$ ?

Ps: This question appeared under bounty for 100 points here https://math.stackexchange.com/questions/4563148/symplectic-form-of-a-coadjoint-orbit-of-a-compact-connected-lie-group but didn't receive any answer so far.

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