What is Borel-de Siebenthal theory? What is Borel-de Siebenthal theory?
 A: I'm not sure the term "theory" is appropriate here, but the joint paper by Borel and de Siebenthal has had considerable influence in Lie theory over the years: MR0032659 (11,326d) 
Borel, A.; De Siebenthal, J.,
Les sous-groupes fermés de rang maximum des groupes de Lie clos.
Comment. Math. Helv. 23, (1949). 200--221.   (There was a short Comptes Rendus announcement in 1948.)   This is found near the start of the Springer four-volume collected papers of Borel, with a couple of minor corrections appended.   
To quote from the review by P.A. Smith:  "Let $G$ be a compact Lie group, $G'$ a closed connected subgroup having the same rank as $G$. Let $Z'$ be the center of $G'$. The main object of this paper is to show that $G'$ is a connected component of the normalizator of $Z'$ in $G$."  The proof involves
"a necessary and sufficient condition that a subsystem of root vectors of $G$ be the root vectors of a closed subgroup of $G$" and the subgroups of this type are found explicitly for all simple $G$.    [Here $G$ is always assumed to be connected.]
The result on subsystems of root systems carries over in a natural way to the study of semisimple complex Lie (or algebraic) groups and their Lie algebras,
for example the determination of subalgebras of maximal rank in the latter.
A: 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.
