What is Borel-de Siebenthal theory?
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.
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.