Since the Fano plane captures certain properties of octonions does it capture certain properties of the $E_8$ root-system?
If so, could you describe how, or provide links to explanations and descriptions (the more elementary, the better).
I am thinking here of the role of the integer 7 in the internal structure of the $E_8$ root-system, e.g. 7*(7+1) = 56, or equivalently, the role of 7 in the internal structure of the associated Coxeter group and its realization in the polytope 4$_2$$_1$.
Please see also this possibly related question at MO: