Basic structure of a Lie algebra
In the case of algebraic structures such as groups and rings, there are basic results about ‘internal’ features (subgroups, subrings) and their relationship with ‘external’ features (such as homomorphisms) that need to be proved before any significant progress can be made. The theory of Lie algebras follows much the same pattern.
3.1 Lie subalgebras
The first necessary definition is that of the analogue of a subgroup or subring.
Definition 3.1.1. A Lie subalgebra (usually just subalgebra) of a Lie algebra is a subspace that is closed under the Lie bracket, i.e. that satisfies
In the previous chapter ...