If group G has a set of generators and set of relations, and given H, K two normal subgroups. Suppose one can write down the elements in H and K explicitly (also in terms of generators and relations). Now how to obtain an element which is in the intersection H\cap K but not in the commutator subgroup [H,K]?
I would just like to mention that this question has topological relevance, as shown in
R. Brown ``Coproducts of crossed $P$-modules: applications to second homotopy groups and to the homology of groups'', Topology 23 (1984) 337-345.
Denote the classifying space of a group $G$ by $BG$. Given normal subgroups $M,N$ of $G$ one can form the space $X$ as the homotopy pushout (i.e. double mapping cylinder) of the two maps $BG \to B(G/M), BG \to B(G/N)$. Then the second homotopy group of $X$ is isomorphic to
$$(M \cap N)/ [M,N] . $$
Actually the result of the paper says more, namely that the homotopy 2-type of $X$ is described by the crossed module $M \circ N \to G$ where $\circ$ is the coproduct of the title of the paper. It is feasible that the question asked could be helped by an analysis of this coproduct. (Note that normal subgroups of a group are special cases of crossed modules.)