Notes on Chow rings of G/B and BG

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 $X$.