Jordan algebras and 3-transposition groups

An idempotent in a Jordan algebra induces a Peirce decomposition of the algebra into subspaces whose pairwise multiplication satisfies a certain fusion rule Φ(12). On the other hand, 3-transposition groups (G,D) can be algebraically characterised as Matsuo algebras Mα(G,D) with idempotents satisfying the fusion rule Φ(α) for some α. We classify the Jordan algebras J which are isomorphic to a Matsuo algebra M12(G,D), in which case (G,D) is a subgroup of the (algebraic) automorphism group of J; the only possibilities are G=Sym(n) and G=32:2. Along the way, we also obtain results about Jordan algebras associated to root systems.