Typically, a model space embedded with a submodel order relationship has a lattice structure, called the model inclusion lattice. Recent works are related to the problems of joint learning of Gaussian graphical models suited for paired data, with exactly two dependent groups of variables. In this framework, it was shown that the model inclusion lattice does not satisfy the distributivity property, and this increases the complexity of procedures for the exploration of the search space. We consider a relevant subfamily of Gaussian graphical models for paired data represented by coloured graphs with common uncoloured structure. We show that this subfamily forms a proper sublattice of the family of Gaussian graphical models for paired data and that, within this sublattice, the distributivity property is satisfied. This can be exploited to improve efficiency in model search procedures.