Axes of Jordan type in non-commutative algebras

The Peirce decomposition of a Jordan algebra with respect to an idempotent is well known. This decomposition was taken one step further and generalized recently by Hall, Rehren and Shpectorov, with their introduction of axial algebras, and in particular primitive axial algebras of Jordan type (PAJs for short). It turns out that these notions are closely related to three-transposition groups and vertex operator algebras. De Medts, Peacock, Shpectorov and M. Van Couwenberghe generalized axial algebras to decomposition algebras which, in particular, are not necessarily commutative. This paper deals with decomposition algebras which are non-commutative versions of PAJs.