  Let $G$ be a compact Lie group and $T$ its maximal torus. The composition of
maps $ H^*(BG)\to H^*(BT) \to 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 $X$.
