In "The topology of torus action on symplectic manifolds" by Michele Audin (1991) p.76 , it is shown that the moment map for the action of a lie group G on a co-adjoint orbit is the inclusion. But I don't understand the proof since the author takes only a vector X of the lie algebra instead of the fondamental vector field associated with X. Can someone explain it to me?

(In the proof, the symplectic form of X and Y at ksi is the duality bracket of ksi and the lie bracket [X,Y]. It is defined in the files that I shared).