TITLE: How to embed $U(1)$ into a bigger group, using Dynkin diagrams
QUESTION [5 upvotes]: 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.

REPLY [5 votes]: "I want to see how the Dynkin diagram of E7 is modified to get A7":
Take $E_7$:
$\bullet -\bullet -\bullet -\stackrel{\stackrel{\textstyle\bullet}{\textstyle |}}{\bullet} -\bullet -\bullet$
Add the lowest negative root (these roots are no longer linearly independent):
$\bullet -\bullet -\bullet -\stackrel{\stackrel{\textstyle\bullet}{\textstyle |}}{\bullet} -\bullet -\bullet -\bullet$
Remove the top root to get another basis of $\mathfrak t^*$:
$\bullet -\bullet -\bullet -\bullet -\bullet -\bullet -\bullet$