$K_{13}$ 3D toroidal cell

Question

Complete graph $K_7$ can be defined by the following toroidal cell acting on the unit triangular lattice.

{{-2,-Sqrt[3]/2},{1/2,Sqrt[3]},{2,Sqrt[3]/2},{-1/2,-Sqrt[3]}}

Here's a picture:

Is there a defining 3D toroidal cell acting on a unit 3D lattice that gives $K_{13}$?