Some remarks on the oscillator group

The structure of the four-dimensional oscillator Lie algebra is examined. The adjoint orbits are determined; these are linearly isomorphic to the coadjoint orbits. The linear subspaces are classified; as a by-product, we arrive at classifications of the full-rank subspaces, the subalgebras, and the ideals. The associated connected Lie groups are then classified. Some comments regarding control systems conclude the paper.