For an arbitrary root α, pick a generator gα of
the root space gα Ď g. Then the family tgαuα is independent, rgαi
, g´αj
s “ 0 for i ‰ j, and
rgα, gβs “ 0 if 0 ‰ α ` β is not a root. Choose a positive root γ of maximal height. Then
rgγ, gαi
s “ rg´γ, g´αi
s “ 0, for all 1 ď i ď r.
If r ą 2, we let Γ be the graph with vertex set V “ t˘α1, . . . , ˘αru and edges tαi
, ´αju for
i ‰ j. Clearly, Γ is connected. Now set ωα “ gα for α P V. Then ω P FpA
.
Γ
,gq is regular and
supppωq “ V, as needed.
If r “ 2 and g ‰ sl2 ˆ sl2, we let V “ t˘α1, ˘α2, ˘γu and declare the edges to be tαi
, ´αju
for i ‰ j and tǫαi
, ǫγu, with ǫ “ ˘1, while if g “ sl2 ˆ sl2, we let V “ t˘α1, ˘α2u and declare
the edges to be tαi
, ´αju for i ‰ j and tǫα1, ǫα2u, with ǫ “ ˘1. In both cases, the resulting
graph Γ is connected, and the desired form ω is constructed as before.
