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

Quadri-algebras, preLie algebras, and the Catalan family of Lie idempotents