Do the 240 roots of E8 implicitly define five disjoint copies of F4 (with 48 roots each)?

Please note that this is an alternative way of formulating the question asked here:

Does a 4_21 exist with 4 vertices from each of of 24 1_22's and 6 from each of 24 "octadeca-diminished" 1_22's (all 48 mutually disjoint)?

and the more general version of this question asked here:

Does the algebraic group E8 ever "collate" two sets of copies of the algebraic group E6?

To see the relationship between this question and those two questions, simply recall that F4 can be obtained from E6 by (Coxeter-Dynkin) diagram folding.

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

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    $\begingroup$ The roots of $E_8$ all have the same length. The roots of $F_4$ don't. $\endgroup$ – Lord Shark the Unknown Nov 29 '17 at 18:49
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    $\begingroup$ You might want to consult math.meta.stackexchange.com/questions/5020/… $\endgroup$ – Lord Shark the Unknown Nov 29 '17 at 18:52
  • $\begingroup$ Thanks for taking the time to respond, and to provide the link. Not sure if your observation (about the different length properties of the roots) rules out the possibility that some intermediate mapping(s) might take the 240 to the five 48's. This is the possibility to which I was trying to allude when I used the word "implicitly" . . . $\endgroup$ – David Halitsky Nov 29 '17 at 18:59
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    $\begingroup$ @DavidHalitsky: Of course with "some intermediate mapping(s)" you can take any set with 240 elements to five sets of 48 elements each, those elements being roots, whales or balloons. The question is whether those mappings respect the given structures, and Lord's comment shows that they would somehow have to change length proportions between non-orthogonal roots, thus cannot be homomorphisms of root systems in the conventional sense. -- There might be something interesting in your observations, but to be frank, I find your presentation in the linked questions nearly impossible to follow. $\endgroup$ – Torsten Schoeneberg Nov 29 '17 at 20:46
  • $\begingroup$ @TorstenSchoeneberg - thanks so much for taking the time to respond - much appreciated. I understand both your points - about "respect" for the given structures, and about the length properties ruling out homomorphisms. I was thinking perhaps of some kind of "Voronoi-like" relation, where the points in each set of 48 are in a certain spatial relationship to a certain 48 of the 240. Of course, I don't mean to imply here that Voronoi cells might be at work here in any formal sense. $\endgroup$ – David Halitsky Nov 29 '17 at 21:13

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