Borel–de Siebenthal theory

Statement

Borel–de Siebenthal theory classifies the closed connected subgroups of maximal rank in a compact connected Lie group by reducing the problem to closed subsystems of its root system. In particular, if $G$ is compact, connected, and simple, the maximal proper connected subgroups $H\subset G$ that contain a maximal torus can be read from the extended Dynkin diagram, with finite central quotients restored at the group level.

Extended-Dynkin-diagram procedure

Choose simple roots $\alpha_1,\ldots,\alpha_r$ , and let

\theta=\sum_{i=1}^r m_i\alpha_i

be the highest root. Adjoin the node $\alpha_0=-\theta$ to obtain the extended Dynkin diagram. The maximal closed subsystems, and hence the semisimple parts of the desired subgroups, arise in two ways. When $m_i=1$ , deleting $\alpha_i$ from the ordinary diagram gives a subsystem of rank $r-1$ , accompanied by a one-dimensional central torus. When $m_i$ is a prime greater than one, replacing $\alpha_i$ by $-\theta$ , equivalently deleting $\alpha_i$ from the extended diagram, gives a proper semisimple subsystem of full rank. These coefficient conditions are what ensure maximality among proper closed connected maximal-rank subgroups.

This procedure is related to, but not identical with, deleting a node from an ordinary Dynkin diagram to obtain a Levi subalgebra. Borel–de Siebenthal theory uses the extended diagram and concerns compact-group subgroups of maximal rank; Levi subalgebras arise from parabolic subalgebras of a complex reductive algebraic or Lie algebra.

Group-level cautions

Root systems determine Lie algebras but not the global isogeny form of a Lie group. Consequently a subgroup identified infinitesimally as $\mathfrak h_1\oplus\mathfrak h_2$ may be globally a finite central quotient of $H_1\times H_2$ . Moreover, “maximal connected subgroup” does not imply maximal among all closed subgroups: a disconnected normalizer may be larger. Both distinctions are essential in applications involving stabilizer identity components.

Example in

F_4

For the compact exceptional group $F_4$ , the theory yields maximal-rank connected subgroups including $\operatorname{Spin}(9)$ and a subgroup with global form $(SU(3)\times SU(3))/\mathbb Z_3$ . In the exceptional Jordan algebra, these appear as identity components of stabilizers of distinguished Jordan subalgebras.

References

Armand Borel and Jean de Siebenthal, “Les sous-groupes fermés de rang maximum des groupes de Lie clos,” Commentarii Mathematici Helvetici 23 (1949), 200–221. Digitized journal record.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026, §§3–4. arXiv:2606.15235.