Note to site moderators: I truly consider this question to be a legit "meta" question about the MathOverflow community - in particular, the general attitude of the MathOverflow community toward a particular tool which is available to them courtesy of HSM Coxeter. So I sincerely ask you not to put this question on hold, nor ask me to delete it.

Background:

Please see my recent question at MathOverflow:

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

Question:

Assume for the sake of discussion that in the examples in the question above:

(a) projection of the vertices of the 8-space cross-polytope onto the vertices of the 3-space octagonal prism DOES preserve all important properties of the roots of $E_8$ (as expressed in their usual {128,112} coodinatization in 8-space)

(b) projection of the vertices of the 9-space cross-polytope onto the vertices of the 3-space nonagonal antiprism DOES preserve all important properties of the roots of $E_6$ (as expressed in 9-space in their usual {18,27,27} coordinatization)

Then under these assumptions, is there still some reason why you personally would NOT think it worthwhile to investigate the roots of $E_6$ and $E_8$ and their relations to one another using the visualizations of these roots in 3-space that can be obtained via Coxeter's projections?

If so, what would your reason be?

For example, would your reason simply arise from your own "personal mathematical aesthetic", e.g a distaste for working from "diagrams" rather than linear symbolic expressions? (I mention this possibility because Dr. (Sam) Eilenberg once chewed me out for doing a 140-page proof about certain properties of the "derivation trees" (ordered finite rooted directed trees with labelled nodes) of derivations in context-free grammars - he told me that in his opinion as an automata-theorist, nothing worthwhile could ever be learned from the properties of "diagrams".)

Or is there some other reason you personally wouldn't use such projections?