Quadratic Jordan algebras whose elements are all invertible or nilpotent

Abstract:

We prove that if $\mathfrak {J}$ is a unital quadratic Jordan algebra whose elements are all either invertible or nilpotent, then modulo the nil radical $\mathfrak {N}$ the algebra $\mathfrak {J}/\mathfrak {N}$ is either a division algebra or the Jordan algebra determined by a traceless quadratic form in characteristic 2. We also show that if $\mathfrak {U}$ is an associative algebra with involution whose symmetric elements are either invertible or nilpotent, then modulo its radical $\mathfrak {U}/\Re$ is a division algebra, a direct sum of anti-isomorphic division algebras, or a split quaternion algebra.