In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type.^[1] There are exactly five of them: ${\displaystyle {\mathfrak {g}}_{2},{\mathfrak {f}}_{4},{\mathfrak {e}}_{6},{\mathfrak {e}}_{7},{\mathfrak {e}}_{8}}$; their respective dimensions are 14, 52, 78, 133, 248.^[2] The corresponding diagrams are:^[3]

• G₂ :
• F₄ :
• E₆ :
• E₇ :
• E₈ :

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

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 ${\displaystyle {\mathfrak {g}}_{2}}$.
• Exceptional Lie algebras may be realized as the derivation algebras of appropriate nonassociative algebras.
• Construct ${\displaystyle {\mathfrak {e}}_{8}}$ first and then find ${\displaystyle {\mathfrak {e}}_{6},{\mathfrak {e}}_{7}}$ as subalgebras.
• Tits has given a uniformed construction of the five exceptional Lie algebras.^[4]