Reduce the non-empty intersection problem on CFGs over $\{a,b\}$ to the set of tuples of four CFGs $G_1,G_2,G_3,G_4$ whose alphabet is $\{a,b\}$ such that $G_1$ and $G_2$ generate at least one common word, $G_3$ and $G_4$ generate at least one common word, but $G_1$ and $G_4$ do not generate a common word, and $G_2$ and $G_3$ do not generate a common word, in order to prove that such set is undecidable (not recursive).