 Dynamical Lie algebra

The dynamical Lie algebra of a finite-dimensional Hamiltonian control system is the real span of its skew-Hermitian drift and control generators and all their nested commutators. The sign choice $iH$ or $-iH$ gives the same algebra. Its connected group determines reachable propagators under the usual freely timed, real-control assumptions. An operator commuting with every generator also commutes with this algebra, imposing invariant sectors.