On Jordan algebras that are factors of Matsuo algebras

Abstract

We describe all finite connected 3-transposition groups whose Matsuo algebras have nontrivial factors that are Jordan algebras. As a corollary, we show that if F is a field of characteristic 0, then there exist infinitely many primitive axial algebras of Jordan type 1/2 over F that are not factors of Matsuo algebras. As an illustrative example, we prove this for an exceptional Jordan algebra over F.