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.



Question about coadjoint orbits of compact connected Lie groups
$\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\ad}{ad}
\DeclareMathOperator{\Lie}{Lie}
\newcommand{\g}{{\mathfrak g}}
\newcommand{\z}{{\mathfrak z}}
\newcommand{\s}{{\mathfrak s}}
\newcommand{\O}{{\mathcal O}}
\newcommand{\wh}{\widehat}
\newcommand{\wt}{\widetilde}$
Let $G$ be a connected compact Lie group, and let $\g$ denote its Lie algebra.
Write $\z$ for the center of $\g$ and set $\s=[\g,\g]$, which is a semisimple Lie algebra.
Then
$$ \g=\z\oplus\s.$$


Write $Z=Z(G)^0$ (the identity component of the center of $G$), $\ \wt S=[G,G]$, $S=G/Z(G)$.
Then $Z$ is a torus, whereas $\wt S$ and $S$ are semisimple Lie groups.
We can identify
$$ \z=\Lie Z,\quad \Lie S=\s=\Lie\wt S.$$


The group $G$ acts on $\g$ by adjoint representation.
Moreover, $G$ acts on $\g$ via the canonical surjective homomorphism
$\pi\colon G\to S$.
We write
$$ g\cdot X=s\cdot X,\quad\text{where} \ g\in G,\ s=\pi(g)\in S,\ X\in\g.$$


We write an element $X\in \g$ as
$$ X=X\_\z+X\_\s\ \quad \text{with}\ X\_\z\in\z,\ X\_\s\in \s\,.$$
Then
$$g\cdot X=X\_\z+g\cdot X\_\s=X\_\z+s\cdot X\_\s\,.$$


Let $\g^\*$ denote the dual space for $\g$.
Then
$$\g^\*=\z^\*\oplus \s^\*.$$
For $r\in \g^\*$ we may write
$$r=r\_\z+r\_\s\quad\text{with}\ r\_\z\in\z^\*, \ r\_\s\in \s^\*.$$
Then for $g\in G$ we have
$$g\cdot r=r\_\z+g\cdot r\_\s=r\_\z+s\cdot r\_\s\,.$$


Let $X\in\g$, $\ X=X\_\z+X\_\s\,$. Then $[X\_\z\,,X\_\s]=0$.
It follows that
$$\exp -tX=(\exp -tX\_\z)\cdot( \exp -tX\_\s) \quad\text{with}\
\exp -tX\_\z\in Z,\ \exp -tX\_\s\in\wt S,$$
whence
$$(\exp -tX)\cdot \alpha=\alpha\_\z+(\exp-t X\_\s)\cdot \alpha\_\s\quad
 \text{for}\ \alpha=\alpha\_\z+\alpha\_\s\in \g^\*.$$
Write
$$ \wh X=\frac d{dt}\Big|\_{t=0}(\exp -tX)\cdot \alpha.$$
Then $\wh X=\wh{X\_\s}$, where
$$ \wh {X\_\s}=\frac d{dt}\Big|\_{t=0}(\exp -tX\_\s)\cdot \alpha\_\s.$$


For $r=r\_\z+r\_\z \in\g^\*$ write $\O\_r=G\cdot r$, $\ \O\_{r\_\s}=S\cdot r\_\s\,$. Then
$$ \O\_r=r\_\z+S\cdot r\_\s=r\_\z+\O\_{r\_\s}\,.$$
Thus for $\alpha=g\cdot r\in \O\_r$ we have
\begin{equation}\label{e:\*}
 T\_\alpha(\O\_r)\cong T\_{\alpha\_\s}(\O\_{r\_\s}).\tag{$\*$}
\end{equation}


Consider the adjoint representation
$$\ad\colon \g\to\mathfrak{gl}(\g),\quad (\ad X)\cdot Y=[X,Y].$$
For $X=X\_\z+X\_\s\in\g$ we have $\ad X=\ad X\_\s\,$.


Consider the Killing form
$$ k\colon \g\times\g\to{\mathbb R}, \quad
(X,Y)\mapsto \Tr\!\big ((\ad X)\cdot (\ad Y)\big).$$
Then
$$ k(X,Y)=\Tr\!\big ((\ad X\_\s)\cdot (\ad Y\_\s)\big)=k(X\_\s\,,Y\_\s).$$


For $\lambda \in\g$, we define a skew-symmetric form on $T\_\alpha(\O\_r)$ by
$$\omega\_\lambda(\wh X,\wh Y)=-k(\lambda,[X,Y]).$$
Since $[X,Y]=[X\_\s\,,Y\_\s]$, for $\lambda=\lambda\_\z+\lambda\_\s$ we have
$$\omega\_\lambda(\wh X,\wh Y)=-k(\lambda,[X,Y])
 =-k(\lambda\_\s,[X\_\s\,,Y\_\s])=\omega\_{\lambda\_\s}(\wh{X\_\s}\,, \wh{ Y\_\s\, }\,).$$


We can identify the tangent spaces
$T\_\alpha(\O\_r)$ and $T\_{\alpha\_\s}(\O\_{r\_\s})$ by \eqref{e:\*}.
Then the formula above is probably what you need.


