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:
and the more general version of this question asked here:
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.