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 \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 exceptional Lie algebras. == References ==