  An idempotent in a Jordan algebra induces a Peirce decomposition of the
algebra into subspaces whose pairwise multiplication satisfies certain fusion
rules $\Phi(\frac{1}{2})$. On the other hand, $3$-transposition groups $(G,D)$
can be algebraically characterised as Matsuo algebras $M_\alpha(G,D)$ with
idempotents satisfying the fusion rules $\Phi(\alpha)$ for some $\alpha$. We
classify the Jordan algebras $J$ which are isomorphic to a Matsuo algebra
$M_{1/2}(G,D)$, in which case $(G,D)$ is a subgroup of the (algebraic)
automorphism group of $J$; the only possibilities are $G =
\operatorname{Sym}(n)$ and $G = 3^2:2$. Along the way, we also obtain results
about Jordan algebras associated to root systems.
