The Powell-Sabin (PS) piecewise quadratic C¹ finite element on the PS 12-split of a triangulation is a common choice for the construction of a BPX-type preconditioner for the biharmonic equation. In this note we investigate the related Powell-Sabin element on the PS 6-split instead of the PS 12-split for the construction of such preconditioners. For the PS 6-split element multilevel spaces can operation, and it alternates the orientation of the refined triangles. Therefore we expect a reduction of the amount of work when compared to the PS 12-split element BPX preconditioner, although both methods have the same asymptotical complexity. Numerical experiments confirm this statement.