# Exceptional Lie algebra

In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: ${\mathfrak {g}}_{2},{\mathfrak {f}}_{4},{\mathfrak {e}}_{6},{\mathfrak {e}}_{7},{\mathfrak {e}}_{8}$ ; their respective dimensions are 14, 52, 78, 133, 248. The corresponding diagrams are:

In contrast, simple Lie algebras that are not exceptional are called classical Lie algebras (there are infinitely many of them).

## 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 of (Fulton & Harris 1991) give a detailed construction of ${\mathfrak {g}}_{2}$ .
• Exceptional Lie algebras may be realized as the derivation algebras of appropriate nonassociative algebras.
• Construct ${\mathfrak {e}}_{8}$  first and then find ${\mathfrak {e}}_{6},{\mathfrak {e}}_{7}$  as subalgebras.
• Tits has given a uniformed construction of the five exception Lie algebras.[citation needed]