  In the paper we describe the subcategory of the category of Z-graded Lie
algebras which is equivalent to the category of Jordan pairs via a functorial
modification of the TKK construction. For instance, we prove that a Z-graded
Lie algebra can be constructed from a Jordan pair if and only if it is
generated by odd graded components and the second graded homology group is
trivial. Similar descriptions are obtained for Jordan triple systems and Jordan
algebras.
