Exceptional Lie algebra

In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional type. There are exactly five of them: ; their respective dimensions are 14, 52, 78, 133, 248. The corresponding diagrams are:

G₂:
F₄:
E₆:
E₇:
E₈:

In contrast, simple Lie algebras that are not exceptional are called classical Lie algebras.

Construction

There is no simple universally accepted way to construct exceptional Lie algebras; in fact, they were discovered only in the process of the classification program. Here are some constructions:

§ 22.1-2 from Fulton and Harris' book give a detailed construction of.
Exceptional Lie algebras may be realized as the derivation algebras of appropriate nonassociative algebras.
Construct first and then find as subalgebras.
Tits has given a uniformed construction of the five exceptional Lie algebras.