Notes on Chow rings of flag varieties G/B and classifying spaces BG
N. Yagita
Let G be a connected compact Lie group and T its maximal torus. The composition of maps
H^{*}(BG) → H^{*}(BT) → H^{*}(G/T)
is zero for positive degree, while it is far from exact.
We change H^{*}(G/T) by Chow ring CH^{*}(X) for
X some twisted form of G/T, and change
H^{*}(BG) by CH^{*}(BG). Then we see that it becomes
near to exact but still not exact, in general. We also
see that the difference for exactness relates
to the generalized Rost motive in X.
Tbilisi Mathematical Journal, Special Issue (7  2021), pp. 529
