  In this paper, we study certain ad-nilpotent subalgebras contained in the
non-zero graded portion of a simple Z_n-graded Lie algebra. These subalgebras
respect the grading on the Lie algebra and are modules for a Borel subalgebra
for the grade-zero Lie subalgebra. We show that semisimple elements in such
subalgebras lie in the center of the subalgebra, and we provide a
classification of these subalgebras whose weight space decompositions have only
non-zero weights.
