Jordan algebra is algebra with multiplication satisfying two axioms (J1) xy=yx (J2) (xxy)x=xx(yx). They were defined in 1934 by Jordan, von Neuman, Wigner seeking for better formalism for quantum mechanics.
In 1966 McCrimmon proposed to analyze instead operator Ux(y)=xyx which lead to a notion of quadratic Jordan algebra. Three axioms of these objects Q1, Q2, Q3 will be written below when proper formatting can be applied.
McCrimmon, Bull. AMS,1978