Root of a Lie algebra

Definition
Given a Lie algebra of dimension $$n$$, a root of the Lie algebra is defined as a weight for the adjoint representation of the Lie algebra.

Root spaces
The root space for a root is the associated weight space, viz the eigenspace for which the eigenvalue works out to be the given root.

Root system
It can be shown that the set of roots of a Lie algebra satisfy the conditions for a root system.