The proof of Brink & Howlett uses minimal (elementary) roots.
One positive root a
dominates another b when the
region b > 0 in the Tits cone is contained
in a > 0. A root is minimal if it dominates only itself.
Bédard's group again.
The number of minimal roots is finite (Brink & Howlett).
When the group is finite, all roots a are minimal.
When affine, minimal roots are the a > 0 and -a + 1
with a for the finite system.