If you have access to Wolfram’s Mathematica, see my notebook with code and data that may be helpful here (135Mb). A PDF is here (65Mb). This has detail on the octonion (E8² and E8³) based constructions for both the 24D Leech lattices as well as the 16D Barnes-Wall lattice shown at the bottom of this post.

The Leech lattice is a sub-quotient of the largest of the sporadic finite simple groups, namely the Monster group. It is related to (and can be constructed by) E₈ and the 23rd Niemeier lattice of E₈³. The 24D Leech lattice is interesting to the physics of sphere packing, error-correcting codes, and possibly unifying General Relativity (GR) with Quantum Mechanics (QM). String theory was founded on the ideas related to its relationship to the Monster group and how 24 dimensions relate to the bosonic energy levels of the partition function on a torus. For more on the Monster and Group Theory, see my post here.

But taking things down a 8D (Bott periodic) notch, roughly speaking the 16D Barnes-Wall lattice is related to E₈ in the same way by BW₁₆=E₈².

I found some very nice recent visualizations on the BW₁₆ WP article by Misaki Ohta with related arXiv papers. He provided links to related code and data here. Below are my renderings using that provided information.

It is interesting to note that if one projects the 61440 2nd shortest 16D vectors of BW₁₆  using orthogonal projection mapping ℝ¹⁶ → ℝ² (i.e. to B₈ basis vectors [x=Cos, y=Sin]@(0-15)π/8 used for E₈ below vs. the proper (0-15)π/16 B₁₆ as done above), we get a similar result as projecting the 2nd shortest 8D vectors of E₈ using the same basis shown as grey vertices in Fig 2 of arXiv:2506.11725:

Projecting the 2160 2nd shortest 8D vectors of E₈ using the B₈ basis vectors including 69120 edges of 8D norm=√8 with colors assigned by projected edge length (notice the similarity in the vertex locations of the figure above):

Projecting the 4320 shortest 16D vectors of BW₁₆  using orthogonal projection mapping ℝ¹⁶ → ℝ² using the B₈ basis vectors [x=Cos, y=Sin]@(0-15)π/8 gives the results shown in Fig. 2 as red and blue vertices:

This time with edges colored by projected edge lengths:

Now, getting creative with the 16D projections of the 4230 BW₁₆, if we double the E8 Petrie basis vectors we get a similar result with 480+1 visible vertices with one at the origin w/240 overlapping vertices. There are 240 yellow with no overlaps and 240 blue with 16 overlaps each. There are 604800 norm=4 edges using roughly the same color pallet as my WP E₈ Petrie image. This makes it easier to see the difference between the 4230 BW₁₆ and the normal E₈ Petrie projection, which seem to share the grid-like pattern of the 24-cells within the folded E₈ = H₄ + φ H₄ :

This is the same as above, but using the 61440 vertex BW₁₆ with no edges (given there are millions of them). Please note that the number of visible vertices is the same as E8₂₄₁ at 2160!

Octonion based Barnes-Wall and Leech Lattice visualizations based on my implementation of work from Geoffrey Dixon and Robert Wilson:

This page presents a comprehensive presentation of the 15 permutations of the six 4D polychora (i.e. 5, 16/8, 24, 600/120 cells). It presents them in full SVG format for convenient use in high quality academic papers on the topic (e.g. simply save the SVG and edit in Inkscape or other tool to produce PDF or PNG, etc.). You are free to use these under Creative Commons Attribution-ShareAlike 4.0 International CC BY-SA 4.0 with appropriate attribution.

Let me know if you need customized hyper-complex and/or hyper-dimensional group-theoretic projection/section visualizations, as my extensive Mathematica code-base may be able to generate it.

The content below uses a 6×15 matrix of 4D to 3D projections of the convex hulls, each with a link to a page with that objects Coxeter section decomposition. I want to give a shout-out (cite) qfbox.info and polytope.miraheze.org for providing vertex coordinates used in the creation of the objects.

There is also another 6×15 table with links to video animations of the sections in the 4D to 3D flatlander from Left/Right (or minus to plus). This is an analog of 3D objects passing through the 2D planar world of flatlanders.

Also available is a Mathematica notebook (360Mb) with code to produce the 4D vertex data as well as the visualized interactive 3D objects. See also an overview of these lists explained in a this Powerpoint presentation, which covers the isomorphisms between them due to the symmetries in the A4, BC4, and H4 Coxeter-Dynkin diagrams that represent them.

Links to the SVG Section Files

Links to Sectioning MP4 Video Animations (i.e. 4D to 3D Flatlander)

For completeness, I am also including a table of 3D polyhedra, namely the Platonic, Archimedean, and Catalan Solids including
their irregular and chiral forms. These were created using
quaternion Weyl orbits directly from the A3, B3, and H3 group
symmetries

.

Another post with more detail on the 5-cell (A4) SU(5)->SU(4) maximal subgroup content generated via quaternions is here.

This is an analysis of the Coxeter Sections for the Swirl Prism 120-Diminished Rectified 600-Cell. The source data, which can be found on either of two polychora sites (qfbox.info and polytope.miraheze.org) seems to have two missing vertices 598 as generated by their permutation lists (vs. the intended 600). The data below, with an added two vertices of {-1,-1,φ³, φ³} and {1,-1,φ³, φ³} in the suspected error sections #3 & #15, is shown in data and 3D convex hull and Coxeter sectioning forms.