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Pre-Jordan algebras were introduced recently in analogy with pre-Liealgebras. A pre-Jordan algebra is a vector space $A$ with a bilinearmultiplication $x \cdot y$ such that the product $x \circ y = x \cdot y + y\cdot x$ endows $A$ with the structure of a Jordan algebra, and the leftmultiplications $L_\cdot(x)\colon y \mapsto x \cdot y$ define a representationof this Jordan algebra on $A$. Equivalently, $x \cdot y$ satisfies thesemultilinear identities: [see PDF]. The pre-Jordan product $x \cdot y = x \succy + y \prec x$ in any dendriform algebra also satisfies these identities. Weuse computational linear algebra based on the representation theory of thesymmetric group to show that every identity of degree $\le 7$ for this productis implied by the identities of degree 4, but that there exist new identitiesof degree 8 which do not follow from those of lower degree. There is anisomorphism of $S_8$-modules between these new identities and the specialidentities for the Jordan diproduct in an associative dialgebra.

Special identities for the pre-Jordan product in the free dendriform algebra