Let $G$ be a connected reductive group over an algebraically closed field and consider a semisimple element $s \in G$ and let $L$ be a Levi subgroup containing $s$.

My question is about the two ways we can look at $C_L(s)^\circ$. On the one hand, it is the connected centralizer of $s$ in the Levi subgroup $L$, on the other hand, it is a Levi subgroup in the connected centralizer $C_G(s)^\circ$.

The Dynkin diagram of $C_{G}(s)^\circ$ can be obtained from the extended Dynkin diagram of $G$ by removing nodes. By removing appropriate nodes from the Dynkin diagram of $C_{G}(s)^\circ$, we obtain the Dynkin diagram of the Levi subgroup $C_{L}(s)^\circ$ of $C_{G}(s)^\circ$.

On the other hand, the Dynkin diagram of $C_L(s)^\circ$ can be obtained from the extended Dynkin diagram of $L$ and the Dynkin diagram of $L$ can be obtained by removing nodes from the Dynkin diagram of $G$.

For example let $G$ have type $B_{10}$ and choose $s$ such that $C_{G}(s)^\circ$ has type $D_4 \times B_6$, then I can find a Levi subgroup of $C_{G}(s)^\circ$ of type $D_4 \times A_2 \times B_3$. This Levi is of the form $C_L(s)^\circ$ for some Levi $L$ of $G$.

However, even though there exists a Levi subgroup of $G$ of type $A_2 \times B_7$ and one of its subgroups of maximal rank has the desired type $A_2 \times D_4 \times B_3$, the bases of the corresponding root systems seem to be completely different to me. In fact, I am not even sure if that subgroup of type $A_2 \times D_4 \times B_3$ lies in $C_G(s)$ or not.

Also, it seems to me that the other Levi subgroups of $G$ do not even have a subgroup of the desired type.

Did I do anything wrong here? If not, why is it not possible for me to obtain the same basis of the Levi of $C_{G}(s)^\circ$ when considering it as a maximal rank subgroup of a Levi subgroup of $G$?

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