  A pre-Lie product is a binary operation whose associator is symmetric in the
last two variables. As a consequence its antisymmetrization is a Lie bracket.
In this paper we study the symmetrization of the pre-Lie product. We show that
it does not satisfy any other universal relation than commutativity. It means
that the map from the free commutative-magmatic algebra to the free pre-Lie
algebra induced by the symmetrization of the pre-Lie product is injective. This
result is in contrast with the associative case, where the symmetrization gives
rise to the notion of Jordan algebra. We first give a self-contained proof.
Then we give a proof which uses the properties of dendriform and duplicial
algebras.
