How to efficiently compute this Pffafian of curvature form

Question

Though one can verify this $$Pf(\Omega) = \frac{1}{8}(\vert \text{Rm} \vert^2 - 4 \vert \text{Ric}\vert^2 + R^2)$$ for a 4-dim manifold by listing out all the curvature components mechanically (I have to ask my computer to do so), is there a clever way to compute this quantity? Thanks