The b ghost of the pure spinor formalism is nilpotent

The ghost for world-sheet reparametrization invariance is not a fundamental field in the pure spinor formalism. It is written as a combination of pure spinor variables which have conformal dimension two and such that it commutes with the BRST operator to give the world-sheet stress tensor. We show that the ghost variable defined in this way is nilpotent since the OPE of b with itself does not have singularities.