Maximal Lie subalgebra

Definition

A Lie subalgebra $\mathfrak m\subsetneq\mathfrak g$ is a maximal Lie subalgebra of $\mathfrak g$ if every Lie subalgebra $\mathfrak k$ satisfying

\mathfrak m\subseteq\mathfrak k\subseteq\mathfrak g

is either $\mathfrak m$ or $\mathfrak g$ . Thus maximality is relative to the chosen embedding and means maximal among all proper Lie subalgebras.

What maximal does not mean

It does not mean “having the greatest possible dimension”; distinct conjugacy classes of maximal subalgebras can have different dimensions.
It does not mean maximal rank. Rank and inclusion maximality are independent conditions.
It does not mean maximal Levi. A maximal Levi is maximal only inside the family of proper Levi subalgebras and is generally contained in a larger proper parabolic subalgebra.
A maximal subalgebra need not be an ideal, so $\mathfrak g/\mathfrak m$ need not inherit a quotient Lie bracket.

Adjoint quotient module

The adjoint action of $\mathfrak m$ preserves $\mathfrak m$ , so it induces a representation on the vector-space quotient $\mathfrak g/\mathfrak m$ :

x\cdot(y+\mathfrak m)=[x,y]+\mathfrak m.

If this $\mathfrak m$ -module is irreducible, then $\mathfrak m$ is maximal, because any intermediate Lie subalgebra would give a nonzero proper invariant subspace. The converse need not hold: an invariant subspace of $\mathfrak g/\mathfrak m$ need not lift to a subalgebra.

Semisimple ambient algebras

For complex simple ambient algebras, maximal subalgebras include regular examples controlled by root data and nonregular examples arising from irreducible representations. Dynkin’s classification organizes these possibilities up to inner automorphism.

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.
Arkady L. Onishchik and Ernest B. Vinberg, eds., Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras, Springer, 1994, Chapter 6. Publisher record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §7. arXiv record.