Free Pre-Lie Algebras are Free as Lie Algebras

We prove that the G-module PreLie is a free Lie algebra in the category of G-modules and can therefore be written as the composition of the G-module Lie with a new G-module X. This implies that free pre-Lie algebras in the category of vector spaces, when considered as Lie algebras, are free on generators that can be described using X. Furthermore, we define a natural filtration on the G-module X. We also obtain a relationship between X and the G-module coming from the anticyclic structure of the PreLie operad.