Quadratic Jordan algebras whose elements are all invertible or nilpotent

Abstract

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