Lie ideal

Summary

An ideal of a Lie algebra $\mathfrak g$ is a linear subspace $\mathfrak a$ with $[x, a] \in \mathfrak a$ for all $x \in \mathfrak g$ and $a \in \mathfrak a$. Ideals are exactly the kernels of Lie algebra homomorphisms, and they are the subspaces by which one can form quotient Lie algebras. They play the role of normal subgroups in group theory. A Lie algebra with no ideals other than $0$ and itself, and not abelian, is called simple.