We compute two small examples separating maximality, maximal transcendence degree and flatness in symmetric Poisson algebras. For the four-dimensional diamond Lie algebra we classify the regular argument shifts whose natural morphisms are flat. The shift at x* is not flat despite satisfying the full codimension-three and invariant-degree hypotheses of the Panyushev-Yakimova maximality theorem. The underlying example and its shift algebra are due to Ooms. For the five-dimensional Heisenberg algebra, the homogeneous cubic F=2z^2t+zp^2-xq^2 has Poisson centralizer k[z,F], a maximal commutative polynomial algebra of transcendence degree two, below the maximal value three. An elementary weighted-degree obstruction for the Airy derivation proves the centralizer formula in all degrees. Ooms's earlier unrestricted negative answer to the maximality/dimension question is explicitly credited. This is an unrefereed manuscript, with AI assistance disclosed in the text and no absolute priority claim.
