I'm having difficulty following computations in the paper "KdV and W-flows" by Zuber.

On pg. 2, what would be the conserved quantity $I_4$, related to the conservation laws of the KdV hierarchy? (Off-chance connection to face polynomials of associahedra.)

On pg. 3, how to interpret the Poisson bracket in Eqn. 2.8 to explicitly derive the RHS? ($u_n$ as defined by Eqn. 2.7 is a scalar, so the bracket makes no sense to me. This is supposedly an analogue to the celebrated Virasoro central extension of the Witt Lie algebra.)