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

Question

Background:

If you look at my recent question 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)?

you will see that this question is one possible narrow formulation of a more general question.

More General Question:

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

If anyone can think of a known case in which E8 does act in this fashion, I would be grateful to know of it.

Thank you for whatever time you can afford to spend considering this matter.