For the coordinates of the symmetric representation of the roots of E6 in 9-space, see section "Roots of E6" here:
Are the roots of E6 in 9-space ever treated as an orthogonal projection of a set of points in 11-space?
If so, please provide a link or reference.
Alternatively, are the roots of E8 ever referred to 11-space? (In which case the subset of these roots comprising the roots of E6 would also be implicitly referred to 11-space.)
Thanks as always for whatever time you can afford to spend considering this question. And please see also this possibly related question: