TITLE: What is Borel-de Siebenthal theory?
QUESTION [14 upvotes]: What is Borel-de Siebenthal theory?

REPLY [11 votes]: In my head at least, part of it is this...
Let G be reductive. Consider the following algorithm:

Extend one component of the Dynkin diagram to its affine diagram, by attaching the lowest root.
From that component, remove one or more vertices. (If you want to stay semisimple, remove only one.)
Repeat to taste. (If you want to stay semisimple, and you're at a disjoint union of $A_m$ diagrams, then you're definitely done.)

The result is the Dynkin diagram of a subgroup H of G of the same rank. Every such subgroup (up to finite factors) arises this way.