$E_6$, $E_8$, and Coxeter's (anti-)prismatic projections of the n-dimensional cross-polytopes

Question

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01/04/2018 - Edited to add: this question was cross-posted at MO here:

https://mathoverflow.net/questions/288114/e-6-e-8-and-coxeters-anti-prismatic-projections-of-the-n-dimensional-cr

and I had to edit the original question in order to address the concerns of some members that the question was not at a "research level" appropriate to MO.

I am therefore curious as to the opinions of MO members regarding the "level" of this question - thanks as always for whatever time anyone can afford to spend considering this matter.

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In Regular Polytopes, Coxeter shows that the vertices of every n-dimensional cross-polytope (hyperoctahedron) project onto the vertices of an n-gonal (anti-)prism.

Has this projection ever been used to visualize properties of $E_8$ in 3-space via the octagonal prism (i.e. by expressing roots in terms of the basis defined by the vectors from the center of the prism to its vertices)

Has this projection ever been used to visualize properties of $E_6$ in 3-space via:

i) the nonagonal antiprism (when the roots of $E_6$ are coordinatized in 9-space)

ii) the octagonal prism (when the roots of $E_6$ are coordinatized in 8-space as a subset of the roots of $E_8$.)

Or are there important properties of $E_6$ and $E_8$ that would not be preserved by such projections?

Please note that this question is related to a comment by Tobias Kildetoft in this question

When an $E_6$ and $E_8$ lattice are co-located by "root co-location", how do mathematicians study the resulting "joint" ($E_6$,$E_8$) neighborhoods?

regarding limitations on his computer-graphic capabilities.

Thanks as always for any time anyone can afford to spend considering this matter.