  Pittie [Pit88] proved that the Dolbeault cohomology of all left-invariant
complex structures on compact Lie groups can be computed by looking at the
Dolbeault cohomology induced on a conveniently chosen maximal torus. We use the
algebraic classification of left-invariant CR structures of maximal rank on
compact Lie groups [CK04] to generalize Pittie's result to left-invariant
Levi-flat CR structures of maximal rank on compact Lie groups.
