Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, homogeneous $G$-variety.

I recently learned that $ X:=F_4/P_4$ is a hyperplane section of $Y:=E_6/P_6$. One obtains the Hasse diagramm of CH(X) by removing one dot in the Hasse diagramm of CH(Y) in degrees $8,12,16$, which is reflected by the fact that the Grothendieck-Chow motive of $Y$ has one more summand $R(8)$, namely a generalized Rost-Motive corresponding to $g_3 \in H^3(k,\mu_3)$ and thus splitting into

$\overline{R(8)} =$ $(\mathbb{Z}$ $ \oplus $ $\mathbb{Z}(4)$ $\oplus $ $\mathbb{Z}(8))(8)$

Question: Are there other examples of algebraic groups $G,H$ with parabolic subgroups $P_i,P_j$ such that $G/P_i$ is a hyperplane section of $H/P_j$ known?