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You might want to note that you need to use a torsion-free connection throughout your calculations.

Yes, indeed my proof relies on the assumption of a connection without torsion.

I think it is also implicit in the text of the exercise: the expression S^[ab NABLA_a S^cd] =0 mentioned by Penrose in the hint to exercise is actually zero only when the connection is without torsion. You can easily verify that: in my solution I shown that it is proportional to the expression in exercise [14.33]; and looking to my solution to [14.33] that again refers to solution to [14.23], you will find that at the end this expression is proportional to the torsion.

Moreover, as per exercise [14.23] the dS itself is proportional to the torsion, so the expression dS=0 is no more an identity, independent from the connection, when there is a torsion.

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Fortunately, the choice of connection doesn't matter to the result, because the Poisson bracket operator only involves 1st derivatives of scalars - and so doesn't involve the connection at all.

I'm not so much convinced by your heuristic argument: the second identity actually involves the second derivative of a scalar. And, in any case, also the expression in the hint by Penrose and the one in exercise [14.33] contain only first order derivatives, however vanish only when there is no torsion.

I really don't know if the second identity is true when there is a torsion.

And because dS=0 is no more an identity I'm not sure that taking into account a torsion in symplectic manifold theory is a trivial generalisation.