Let A and B be point lattices of dimensions m and n, respectively. The intersection of A and B is a lattice C of dimension k. A recently proposed lattice construction technique allows one to construct a lattice D by gluing A and B along C. The technique has furnished new lattices of record center densities, but it is limited to the cases where k equals 1 or 2. In the original work the generator matrices of the new lattices were not presented in explicit form. The purpose of this work was to display those generator matrices in explicit form and investigate the case where k equals 3.
