Removing a simple root

Core idea

Let $\mathfrak g$ be a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak h$ , root system $\Phi$ , and simple roots $\Delta$ . To remove the simple root $\alpha_0\in\Delta$ , set

I=\Delta\setminus\{\alpha_0\}, \qquad \Phi_I=\Phi\cap\operatorname{span}_{\mathbb Z}(I).

The set $\Phi_I$ is the full root subsystem whose Dynkin diagram is obtained by deleting the vertex $\alpha_0$ and its incident edges.

Two associated subalgebras

Let $\mathfrak h_I$ be the span in $\mathfrak h$ of the coroots $h_\alpha=[e_\alpha,e_{-\alpha}]$ for $\alpha\in I$ . Deleting $\alpha_0$ gives the regular semisimple subalgebra

\mathfrak g_I =\mathfrak h_I\oplus\bigoplus_{\beta\in\Phi_I}\mathfrak g_\beta.

Its rank is one less than that of $\mathfrak g$ .

Retaining the whole Cartan instead gives

\mathfrak l_I =\mathfrak h\oplus\bigoplus_{\beta\in\Phi_I}\mathfrak g_\beta.

This is a maximal Levi subalgebra, with

\mathfrak l_I=\mathfrak g_I\oplus Z(\mathfrak l_I)

and one-dimensional center.

Caution about the Cartan part

The Cartan subalgebra of the semisimple algebra $\mathfrak g_I$ is the span of the retained coroots. It is generally not the kernel of the deleted simple root $\alpha_0:\mathfrak h\to\mathbb C$ . The coroot span is what ensures that brackets $[\mathfrak g_\beta,\mathfrak g_{-\beta}]$ remain inside $\mathfrak g_I$ .

Iteration

Removing several simple roots amounts to choosing a smaller subset $I\subseteq\Delta$ . Iterating the construction produces chains of regular subalgebras. For example, suitable successive deletions give the exceptional chain

A_2\sqcup A_1\subset A_4\subset D_5\subset E_6\subset E_7.

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§8, 14. Publisher record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §2. arXiv record.