A separable L-embedded Banach space has property (X) and is therefore   the unique predual of its dual
  In this note the following is proved. Separable L-embedded spaces - that is separable Banach spaces which are complemented in their biduals such that the norm between the two complementary subspaces is additive - have property (X) which, by a result of Godefroy and Talagrand, entails uniqueness of the space as a predual. 
Hermann Pfitzner (MAPMO)