  We compute the expansion of the Catalan family of Lie idempotents introduced
in [Menous et al., Adv. Applied Math. 51 (2013), 177-22] on the PBW basis of
the Lie module. It is found that the coefficient of a tree depends only on its
number of left and right internal edges. In particular, the Catalan idempotents
belong to a preLie algebra based on naked binary trees, of which we identify
several Lie and preLie subalgebras.
