Quadratic Jordan algebras whose elements are all regular or nilpotent

Abstract:

We prove that if $J$ is a quadratic Jordan algebra whose elements are all either regular or nilpotent, and which satisfies a common multiple property (that whenever $z$ is nilpotent and $\upsilon$ regular then $\operatorname {Im} {U_\upsilon } \cap \operatorname {Ker} {U_z} \ne 0$), then modulo the radical $R$ the algebra $J/R$ is either a domain or a form of a Jordan algebra determined by a traceless quadratic form in characteristic 2.