The canonical Hamiltonian bracket satisfies the Lie algebra laws

For every natural number $l$, the bilinear extension of the canonical Hamiltonian basis brackets is a complex Lie bracket. Write $h_0=0$, let nonzero exponent vectors index the remaining $h_r$, and let $d_i$ denote the degree generators. The brackets are

with $[h_r,d_i]=-r_i h_r$. Here $\widetilde r$ exchanges the two coordinate blocks and negates the second block. The assertion is precisely additivity and complex homogeneity in both variables, alternation, and the cyclic Jacobi identity for the explicitly defined bracket on the finitely supported vector-space carrier. It supplies the bracket-law clause needed by the canonical Hamiltonian application.