Cross-pair cancellation need not diagonalize a bosonic Hamiltonian

For a two-mode quadratic Hamiltonian, a transformation can remove $AB+A^{†}{B}^{†}$ while leaving $A^{†2}$ or $B^{†2}$ terms. Such terms create quanta from the proposed vacuum, so their absence must also be checked before identifying that vacuum with an energy eigenstate. Coordinate coupling $kxy$ contains both pair and number-conserving terms and supplies an explicit example.