In exercise 5.3-4. in Marsden's book I'm asked to
prove that $\mathbf d \Omega^{fs} = 0$ on $\mathbb P \mathcal H$ directly,
where $\mathbb P \mathcal H$ is an arbitrary projective Hilbert space (possibly infinite dimensional) endowed with the fubini-study form $\Omega^{fs}$. He already proved this using the invariance of $\Omega^{fs}$ under the action of $\mathbb P U(\mathcal H)$, which I'm not allowed to use.
The symplectic form on $\mathcal H$ is given as $$\Omega(\varphi_1, \varphi_2) := -2 \, \mathrm {Im} \, \left<\varphi_1, \varphi_2 \right>,$$and there is also the natural projection $$\pi: \mathcal H \setminus \{0\} \to \mathbb P \mathcal H,\; \psi \mapsto [\psi] := \mathbb C \psi,$$ with its associated tangent map $$T\pi: \left\{ \begin{align} &\mathcal T(\mathcal H \setminus \{0\}) &&\to T(\mathbb P \mathcal H)\\ &\varphi \in \mathcal T_{\psi}(\mathcal H \setminus \{0\}) &&\mapsto \varphi + [\psi] \in T_{[\psi]}(\mathbb P \mathcal H) \cong \mathcal H \,/\, \mathbb C \psi. \end{align}\right.$$
On p. 155 in Marsden's book the Fubini study form on $\mathbb P \mathcal H$ was defined in terms of the symplectic form on $\mathcal H$ as follows:
For $[\psi] \in \mathbb P \mathcal H$, $\|\psi\| = 1,$ and $\varphi_1, \varphi_2 \in (\mathbb C \psi)^{\bot}$, $$\Omega^{fs}_{[\psi]}(T_{\psi}\pi(\varphi_1), T_{\psi}\pi(\varphi_2)) := \Omega(\varphi_1, \varphi_2).$$
This looks like a pushforward under the natural projection $\pi_{\mathcal S} : \mathcal S \to \mathbb P \mathcal H$, from the unit sphere $\mathcal S = \{ \psi \in \mathcal H \mid \|\psi\|^2 = 1\}$ onto $\mathbb P \mathcal H$, i.e. $\Omega^{fs} = {\pi_{\mathcal S}}_* \Omega$. But of course it isn't since $\pi_{\mathcal S}$ is not injective. And hence ${\pi_{\mathcal S}}_* \Omega$ is not even defined (under the inverse image of $T\pi$ both arguments might end up in different tangent spaces). That's a real bummer because I already managed to show that $\mathbf d \Omega = 0$. Using this result I now just need a justification so that I can say
For $[\psi] \in \mathbb P \mathcal H$, $\|\psi\| = 1,$ and $\varphi_1, \varphi_2, \varphi_3 \in (\mathbb C \psi)^{\bot}$, $$\mathbf d \Omega^{fs}_{[\psi]}(T_{\psi}\pi(\varphi_1), T_{\psi}\pi(\varphi_2), T_{\psi}\pi(\varphi_3)) = \mathbf d \Omega(\varphi_1, \varphi_2, \varphi_3).$$
But how can I do that? Thanks in advance for any hints / suggestions.