  We prove 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. We also show
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 (non
necessarily continuous) biorthogonality preserving linear surjection between
them.
