Equality of B(Jord(V)) and Inner(Jord(V)) for free Jordan algebras

Establish whether, for every vector space V over a field of characteristic zero, the functorial space B(Jord(V)) is isomorphic to the Lie algebra of inner derivations Inner(Jord(V)) of the free Jordan algebra Jord(V); that is, determine whether B(Jord(V)) ≅ Inner(Jord(V)) holds for all free Jordan algebras.

Background

The paper studies the Tits–Allison–Gao (TAG) construction and related conjectures of Kashuba and Mathieu concerning free Jordan algebras. In this framework, B(J) is the functorial replacement for the space of inner derivations Inner(J), and the Kashuba–Mathieu conjecture would imply B(Jord(V)) ≅ Inner(Jord(V)).

While the authors disprove the central conjecture, they note that B( SJord(V) ) ≅ Inner( SJord(V) ) for free special Jordan algebras, and that Jord(V) ≅ SJord(V) when dim(V)=2, giving B(Jord(V)) ≅ Inner(Jord(V)) in that case. This motivates asking whether the equality persists for all free Jordan algebras despite the failure of the conjecture.

References

This raises a natural question whether we always have B(\Jord(V))\cong\mathrm{Inner}(\Jord(V)) even though it cannot be deduced from the (false) Conjecture \ref{conj:conj1}.

— On the conjecture of Kashuba and Mathieu about free Jordan algebras  (2507.00437 - Dotsenko et al., 1 Jul 2025) in Section 4, Subsection “Inner derivations of free Jordan algebras”