In this paper, we demonstrate that the dynamics of an n-dimensional Lindblad control system can be separated into its inter- and intraorbit dynamics when there is fast controllability. This can be viewed as a control system on the simplex of density operator spectra, where projectors representing the eigenspaces are viewed as control variables. The local controllability properties of this control system can be analyzed when the control set of projectors is limited to a finite subset. In particular, there is a natural finite subset of n! projector tuples that are effective for low-purity orbits.