The concept was introduced by Springer (1973) to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation.
Over fields of characteristic not equal to 2, the theory of J-structures is essentially the same as that of Jordan algebras.
A J-structure is a triple (V,j,e) where V is a vector space over K, j is a birational map from V to itself and e is a non-zero element of V satisfying the following conditions.
Let Q be the usual sum of squares quadratic function on Kr for fixed integer r, equipped with the standard basis e1,...,er.
[5] In characteristic not equal to 2, which we assume in this section, the theory of J-structures is essentially the same as that of Jordan algebras.
In general characteristic, which we assume in this section, J-structures are related to quadratic Jordan algebras.
In the opposite direction, if (V,Q,e) is a separable quadratic Jordan algebra with inversion i, then (V,i,e) is a J-structure.
[12] McCrimmon proposed a notion of H-structure by dropping the density axiom and strengthening the third (a form of Hua's identity) to hold in all isotopes.