Ideal in a Lie algebra

Let $\mathfrak g$ be a Lie algebra .

Definition (Ideal).
A linear subspace $\mathfrak i\subseteq \mathfrak g$ is an ideal if it is a Lie subalgebra and

Equivalently, $\mathfrak i$ is stable under the adjoint action : for every $x\in\mathfrak g$, the endomorphism $\mathrm{ad}_x$ maps $\mathfrak i$ into itself.

Basic consequences.

• If $\phi:\mathfrak g\to\mathfrak h$ is a Lie algebra homomorphism , then $\ker\phi$ is an ideal.
• If $\mathfrak i$ is an ideal, the quotient vector space $\mathfrak g/\mathfrak i$ carries a natural quotient Lie algebra structure.

Examples.
The center $Z(\mathfrak g)$ is an ideal, and the derived subalgebra $[\mathfrak g,\mathfrak g]$ is an ideal (see the ideal lemma ).

Context.
Ideals play the role of normal subgroups in Lie theory and are central in structure results such as the Levi decomposition .