I am trying to find the embedding and the branching rules for some group decompositions. For example, I consider $E_7$ and its maximally compact subgroup $SU(8)$ and I want to "see" how the Dynkin diagram of $E_7$ is modified to get $A_7$. I tried to take a linear combination of roots to produce $A_7$. And also following other analog questions, with this I have (almost) no issues.

The problem is for an embedding like $SU(8)⊃SU(6)×U(1)$. How do I get this? As all of you can imagine, my doubts arise when the embedding involves abelian subgroups, that I am not able to see from Dynkin diagrams.

All these issues comes from the calculus of centraliser groups. For instance, I want the centraliser group $\mathcal{C}_{E_7}(SU(3))$ of $SU(3)_{\mathrm{diag}} \subset SU(3) \times SU(3) \subset SU(6) \subset SU(8) \subset E_7$. I know it is $$ \mathcal{C}_{E_7}(SU(3)) = SU(3) \times SU(2) $$ but I do not have a procedure to get something involving $U(1)$ factors.

1more comment