Let $\Phi$ be an irreducible root system. A root subsystem of $\Phi$ is a subset $\Psi \subseteq \Phi$ which is a root system. One can find the possible types of root subsystems of $\Phi$ by deleting nodes in the extended Dynkin diagram of $\Phi$.
Now suppose that we fix a possible type $T$ (e.g. $A_2$) of root subsystems of $\Phi$. What I'm interested in is the following:
- How many root subsystems of type $T$ are there in $\Phi$? One could, of course, brute force this using a computer but I guess there is some kind of formula or more elaborate method?
- How do I find all root subsystems of type $T$ in $R$? Again, one could just generate all possible root subsystems and check their type but is there something smarter one could do?
Is there any reference for these types of questions? Has anyone read Carter's Conjugacy classes in the Weyl group and know it well enough to tell if I will find my answers there (I don't know this area well enough to quickly tell by browsing his paper)?
By the way, I'm especially interested in the case of root subsystems of type $D_2$ in $D_5$ and the case of root subsystems of type $A_2 \times A_2 \times A_2$ in $E_6$.