Maximal connected closed subgroup

Definition

Let $G$ be a Lie group. A maximal connected closed subgroup is a proper subgroup $H<G$ that is connected and closed, and for which every connected closed subgroup $K$ satisfying

H\subseteq K\subseteq G

is either $H$ or $G$ .

What maximality does not say

The adjective “connected” restricts the competitors $K$ . Thus $H$ can be maximal among proper connected closed subgroups while lying inside a proper disconnected closed subgroup of $G$ . Consequently, “maximal connected closed subgroup” is weaker than “maximal closed subgroup.”

The term also does not mean merely the identity component of some maximal subgroup; it is a maximality condition in its own right.

Example in compact

F_4

For the compact exceptional group $F_4$ , subgroups isomorphic to

\operatorname{Spin}(9) \qquad\text{and}\qquad \bigl(SU(3)\times SU(3)\bigr)/\mathbb Z_3

occur as maximal connected closed subgroups. In the exceptional-Jordan-algebra model, the second is the identity component of the stabilizer of a copy of $\mathfrak h_3(\mathbb C)$ . Its full stabilizer is disconnected, illustrating why connected maximality and unrestricted closed maximality differ.

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. EuDML record.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv record. Relevant: §3.