Extend triangular grid to 3D space?

My interpretation of the close-neighbours-neighbouring property is as follows:

For each node, the minimum separation among all distinct pairs of its neighbours is the same as the distance between the node and its neighbours.

Then the fcc lattice (right side of the diagram below) satisfies the two properties, having 12 neighbours per node. The spaces between the nodes then form the alternated cubic/tetrahedral–octahedral honeycomb.