Special elements of a lattice $L$ (distributive, codistributive, neutral etc.) induce a congruence relation on $L$. Here we consider the following problem: If the lattice identity is satisfied on a class of that congruence (a sublattice of $L$), under which conditions this identity holds on the lattice itself? Several algebraic results are deduced from the obtained lattice properties: characterizations of the congruence extension and of the congruence intersection properties, and some general properties of the weak congruence lattice of an algebra.