A Jordan algebra is an algebra with multiplication satisfying two axioms
(J1) xy=yx
(J2) (xxy)x=xx(yx).
They were defined in 1934 by Jordan, von Neumann, and Wigner seeking a better formalism for quantum mechanics.

In 1966 McCrimmon proposed to analyze instead the operator Ux(y)=xyx, which lead to a notion of quadratic Jordan algebras. Three axioms (Q1, Q2, Q3) of these objects can be found below.