Are the roots of E6 in 9-space ever treated as an orthogonal projection of a set of points in 11-space?

Question

Background:

For the coordinates of the symmetric representation of the roots of E6 in 9-space, see section "Roots of E6" here:

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

Question:

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:

Cases of (partly) "coincident" orthogonal projections of n-dimenstional polytopes in (n-k)-spaces