This paper proves that every biorthogonality preserving linear surjection from a weakly compact JB*-triple containing no infinite dimensional rank-one summands onto another JB*-triple is automatically continuous. It also shows that every biorthogonality preserving linear surjection between atomic JBW*-triples containing no infinite dimensional rank-one summands is automatically continuous. Consequently, two atomic JBW*-triples containing no rank-one summands are isomorphic if and only if there exists a biorthogonality preserving linear surjection between them. The paper discusses the properties of JB*-triples, including their structure, annihilators, and the concept of biorthogonality preserving linear maps. It also provides a detailed analysis of the automatic continuity of such maps in the context of JB*-triples and JBW*-triples. The results are applied to show that certain types of JB*-triples are isomorphic if there exists a biorthogonality preserving linear surjection between them. The paper also includes a detailed study of the structure of JB*-triples, including their decomposition into Peirce components, and the properties of minimal tripotents and their associated inner ideals.This paper proves that every biorthogonality preserving linear surjection from a weakly compact JB*-triple containing no infinite dimensional rank-one summands onto another JB*-triple is automatically continuous. It also shows that every biorthogonality preserving linear surjection between atomic JBW*-triples containing no infinite dimensional rank-one summands is automatically continuous. Consequently, two atomic JBW*-triples containing no rank-one summands are isomorphic if and only if there exists a biorthogonality preserving linear surjection between them. The paper discusses the properties of JB*-triples, including their structure, annihilators, and the concept of biorthogonality preserving linear maps. It also provides a detailed analysis of the automatic continuity of such maps in the context of JB*-triples and JBW*-triples. The results are applied to show that certain types of JB*-triples are isomorphic if there exists a biorthogonality preserving linear surjection between them. The paper also includes a detailed study of the structure of JB*-triples, including their decomposition into Peirce components, and the properties of minimal tripotents and their associated inner ideals.