  We investigate the factorization problem as well as the classifying
complements problem in the setting of Jordan algebras. Matched pairs of Jordan
algebras and the corresponding bicrossed products are introduced. It is shown
that any Jordan algebra which factorizes through two given Jordan algebras is
isomorphic to a bicrossed product associated to a certain matched pair between
the same two Jordan algebras. Furthermore, a new type of deformation of a
Jordan algebra is proposed as the main step towards solving the classifying
complements problem.
