Reference request: Projective representations of a simply connected real semisimple Lie group lift to unitary representations I recently got interested in representation theory in quantum mechanics and I read the following theorem:

Let $G$ be a simply-connected Lie group with $H^2(\mathfrak{g},\mathbb{R})=0$ and let $\mathcal{H}$ be a complex Hilbert space. Then every projective representation $\rho:G\to \text{Aut}(\mathbb{P}(\mathcal{H}))$ lifts to a unitary representation $\pi:G\to U(\mathcal{H})$.

I am looking for a proof of the theorem above, does anyone have a reference where it is proven?
 A: The original reference is Thms 3.2 and 7.1 in
Bargmann, V., On unitary ray representations of continuous groups, Ann. Math. (2) 59, 1-46 (1954). ZBL0055.10304. 
Added: For a nice shortened proof, I just noticed this very commendable paper (see linked review):
Simms, D. J., A short proof of Bargmann’s criterion for the lifting of projective representations of Lie groups, Rep. Math. Phys. 2, 283-287 (1971). ZBL0232.22021.
A: Since $H^2(\mathfrak{g},\mathbb{R})=0$ and since $G$ is simply connected, it follows from the van Est theorems that $H^2(G,S^1)=0,$ which means that all $S^1$- extensions of $G$ are trivial. But a lifting of a projective representation $\rho$ of $G$ is equivalent to the data of a trivialization of the $S^1$ - extension of $G$ given by the pullback (by $\rho$) of the extension $1\to S^1\to U(\mathcal{H})\to \mathbb{P}(U(\mathcal{H}))\to 1,$ which from the above van Est argument must exist.
For the relevant van Est theorems, see https://mathscinet.ams.org/mathscinet-getitem?mr=59285
or Theorem 2.3 in
https://arxiv.org/abs/1909.12100
Edit: In the finite dimensional case the Lie algebra extension associated to the extension  $1\to S^1\to U(\mathcal{H})\to \mathbb{P}(U(\mathcal{H}))\to 1$ splits, and the proof I gave can be tweaked to show that projective unitary representations of simply connected Lie groups always lift to unitary representations. It probably doesn't split in the case that the hilbert space is infinite dimensional and this is why the assumption that  $H^2(\mathfrak{g},\mathbb{R})=0$ is needed. In this case there is a slight gap in my proof since I didn't show that the pullback of the extension is actually a smooth $S^1$ - extension of $G$, but maybe this isn't hard to show.
