7-limit lattice

Ran across the basis for the 7-limit lattice on page 34 of
SPLAG (Sphere Packing Lattices and Groups, Conway et al) using
the cubeoctohedron and Voronoi cells, etc. So I am committed to
getting this down so I will be able to be contribute questions
w.r.t. lattices.

Just one thought, using D4, (or the M5 symmetry), (1,7)(3,9)(5,11)
generators, one can see that 3/2-> 21/20 using 7/10 and 3/2-> 15/14
using 5/7, and 3/2 -> 35/24 using 35/36 comma. Is this just a
coincidence? These are the coordinates of the seven-limit lattice...

Just my hunch that S3, D4 and the 7-limit are intimately connected
in 12-EDO. S4 doesn't seem very meaningful though. Also, 35/24 also
approximates 1/(ln(2)) in the trapezoid approximation of the integral
of 1/x between 1 and 2 when split in two parts, another coincidence?

Okay, this will be my last trigger-happy post (for awhile). Lot of
thoughts to get together but not much time, I am afraid.

PH