Centralizer of a regular reductive subalgebra

Statement

Let $\mathfrak g$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak h$ and root-space decomposition

\mathfrak g=\mathfrak h\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha.

Suppose $\mathfrak k\subseteq\mathfrak g$ is regular and reductive in $\mathfrak g$ , so that it can be written

\mathfrak k=\mathfrak t\oplus\bigoplus_{\alpha\in\Psi}\mathfrak g_\alpha, \qquad \mathfrak t=\mathfrak k\cap\mathfrak h,

with $\Psi=-\Psi$ and with $\mathfrak t$ containing the coroots of $\Psi$ . Then its centralizer is

C_{\mathfrak g}(\mathfrak k) =\mathfrak t_0\oplus\bigoplus_{\beta\in\Omega}\mathfrak g_\beta,

where

\mathfrak t_0 =\{H\in\mathfrak h:\alpha(H)=0\text{ for all }\alpha\in\Psi\}

and $\Omega$ consists of the roots $\beta\in\Phi$ satisfying

\beta|_{\mathfrak t}=0 \quad\text{and}\quad [\mathfrak g_\beta,\mathfrak g_\alpha]=0 \text{ for every }\alpha\in\Psi.

Simply laced simplification

Assume $\Phi$ is simply laced. Because $\mathfrak t$ contains the coroots of $\Psi$ , the condition $\beta|_{\mathfrak t}=0$ makes $\beta$ orthogonal to every root in $\Psi$ . In a simply laced system this already implies $\beta\pm\alpha\notin\Phi$ for all $\alpha\in\Psi$ . Hence

\Omega=\{\beta\in\Phi:\beta|_{\mathfrak t}=0\}.

Using the Killing form to identify $\mathfrak h$ and $\mathfrak h^*$ , this says: retain the Cartan directions orthogonal to the root span of $\mathfrak k$ , and retain exactly those ambient root spaces whose roots are orthogonal to the entire toral part $\mathfrak t$ .

Why the toral part matters

If $\mathfrak k$ has a nontrivial center, then $\mathfrak t$ can be strictly larger than the span of its coroots. A root can be orthogonal to every root of $\mathfrak k$ while failing to vanish on these extra central directions. Such a root space does not centralize $\mathfrak k$ . Thus replacing $\beta|_{\mathfrak t}=0$ merely by orthogonality to $\Psi$ is not valid without an additional hypothesis.

General root-system caution

The bracket condition in the definition of $\Omega$ is the general criterion. In a crystallographic root system, once $\beta$ vanishes on $\mathfrak t$ and $\mathfrak t$ contains the coroots of $\Psi$ , orthogonality to every $\alpha\in\Psi$ forces $\beta\pm\alpha\notin\Phi$ by the root-string property, so the simply-laced simplification extends to this setting. Orthogonality to $\Psi$ alone remains insufficient when $\mathfrak t$ has extra central directions.

References

Eugene B. Dynkin, “Semisimple subalgebras of semisimple Lie algebras,” Matematicheskii Sbornik 30(72), no. 2 (1952), 349–462; English translation, AMS Translations, Series 2, vol. 6 (1957), 111–244. Journal record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §§3, 6–7. arXiv record.