Poisson bracket from a central deformation parameter ($\{\bar{a},\bar{b}\}=h^{-1}[a,b]\mathrm{mod}h$)

Let $h$ be central and a non-zero-divisor in an associative algebra, with commutative quotient modulo $h$. Dividing commutators by $h$ and reducing modulo $h$ is independent of the lifts, since changing a lift by $hc$ changes the quotient by a commutator, zero modulo $h$. Antisymmetry, the Leibniz rule and the Jacobi identity descend from the corresponding commutator identities. For the order filtration of a Weyl algebra, the convention is $\{{\xi }_i,x_j\}={\delta }_{ij}$, with the opposite sign for $\{x_i,{\xi }_j\}$.