My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
canonical G-invariant symplectic structure. 
Kirillov defined the coadjoint orbit by the natural way as follows,


Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra,and also $\mathfrak{g^\*}$ be the dual of Lie algebra


$\mathfrak{G}=\{Ad^\*(g)F, g\in G\}$ where $F\in\mathfrak{g^\*}$.


In fact, Kirillov introduced an antisymmetric bilinear form $B\_F$ on $\mathfrak{g}$ by


$B\_F(X,Y)=<F,[X,Y]>$ and showed that $B\_F$ is invariant under $Stab(F)=\{g\in G: Ad^\*(g)F=F \}$ and by using this fact, he introduced a $G$-Invariant symplectic structure $\omega\_{\mathfrak{G}}(F)(ad^\*(X)F,ad^\*(Y)F)=B\_F(X,Y)$ on $\mathfrak{G}$ which is now known as Kirillov-Kostant-Souriau Theorem .


Now, I am trying to extend this result for $\mathfrak{g}\oplus \mathfrak{g^\*} $ instead of $\mathfrak{g^\*} $and try to find a G-invariant symplectic structure. 
The fact is that $\mathfrak{g}\oplus \mathfrak{g^\*}$ is exactly "equal"(in algebraic and geometric sense) to $Lie(T^\*G)$, and we have a symplectic structure on $T^\*G$ so we will have a symplectic structure on $Lie(T^\*G)$, because we just can restrict the symplectic structure of $(T^\*G)$ to $T\_e(T^\*G)$ and get a symplectic structure on $Lie(T^\*G)$.(Also we can define a bilinear symmetric and antisymmetric form on $\mathfrak{g}\oplus \mathfrak{g^\*}$ and define a generalized complex structure on it and so we will have a symplectic, poisson, kahler structure on $\mathfrak{g}\oplus \mathfrak{g^\*}$ by this way. )
But it would be a good question , if we define the action of $g\in G$ on $X+F\in \mathfrak{g}\oplus \mathfrak{g^\*} $ by $g.(X+F)=Ad(g)X+Ad^\*(g)F$.
and define an orbit as same method of kirillov by $\mathfrak{O}=\{Ad(g)X+Ad^\*(g)F, g\in G\}$.


So, by this definition,we will have $Stab(X+F)=\{g\in G: Ad^\*(g)F=F , gX=Xg \}$, and then $G$ is a fibre bundle over the base $\mathfrak{O}$.


Now, if we construct the symplectic structure


$\omega\_{X+F}:T\_{X+F}{\mathfrak{O}}\times T\_{X+F}{\mathfrak{O}}\to R$


$\omega\_{X+F}([Y,X]+ad^\*(Y)F,[Z,X]+ad^\*(Z)F)$=?
THE FACT IS "?" (I mean right hand sight of equality) should be an invariant bilinear form $B\_{X+F}(Y,Z)$ and we should find it. So, my question is how can we find this guy?


PS:I edited my question after comments of Ben and Mariano .



Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $
There is no hope for such a construction. The orbits of $G$ on $\frak{g}\oplus\frak{g}^\ast$ are not always of even dimension, so they will not support symplectic structures. 


For example, take $G=\mathrm{SO}(3)$. Because $G$ is simple, $\frak{g}^\ast$ is isomorphic to $\frak{g}$ as a $G$-module, so the action of $G$ on $\frak{g}\oplus\frak{g}^\ast$ is just the diagonal action of $G$ on $\frak{g}\oplus\frak{g}$. Now, if $x, y\in\frak{g}$ are linearly independent, then the stabilizer of $(x,y)\in \frak{g}\oplus\frak{g}$ is trivial, so its orbit is just a copy of $G$, which has dimension $3$.


A similar phenomenon happens for any odd-dimensional compact simple Lie group.


