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Consider the 72 roots of the algebraic group E6 in their most symmetric coordinatization (in 9-space), as given in the section "Roots of E6" here:

https://en.wikipedia.org/wiki/E6_(mathematics)

From these roots, we can clearly form ordered palindromic triples of roots RiRjRk where Ri is the same root as Rk.

Are such ordered palindromic triples of roots of any particular interest in E6 or structures derived from E6?

If so, where and why?

Thanks for whatever time you can afford to spend considering this question.

David Halitsky
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  • This is very similar to https://math.stackexchange.com/questions/2504671/why-do-the-72-roots-of-the-algebraic-group-e6-all-have-internal-triplet-struct – Angina Seng Nov 14 '17 at 17:10
  • @LordSharktheUnknown - thanks for taking the time to respond. But forgive me for not understanding what you mean by "similar". The prior question (about triplet structure of the coordinate 9-tuples) involves properties of a particular coordinatization of the roots of E6, On the other hand, this question involves certain ordered triples of roots, REGARDLESS of the coordinatization we choose for these roots (the 9-space symmetric coordination, or Coxeter's original coordinatization in 8-space, or the Coxeter/Conway coordinatization in 6-space.) Could I ask you to clarify your comment ? – David Halitsky Nov 14 '17 at 17:24
  • In the above comment, I should not have written "the 9-space symmetric coordination, or Coxeter's original coordinatization in 8-space, or the Coxeter/Conway coordinatization in 6-space." Instead, I should have written "the 9-space symmetric coordination, or an 8-space coordinatization a la Coxeter, or a 6-space coordinatization a la Coxeter/Conway". My apologies for the bad wording. – David Halitsky Nov 14 '17 at 17:32
  • Related question 11/15/2017: 02:25 US Eastern DST: any role in algebraic group E6 for pairs of ordered triples of roots ((RiRjRk), (RuRvRw)) such that Ri = Rw, Rk=Ru, and Rj=Rv ? (So that the 6-tuple (RiRjRkRuRvRw) is itself palindromic . . . ) – David Halitsky Nov 15 '17 at 19:25

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