Ideal of a Lie algebra

A vector subspace $I\subseteq \mathfrak{g}$ is an ideal when $[\mathfrak{g},I]\subseteq I$. It is therefore the kernel of a Lie-algebra quotient map.

For a Lie algebra ideal $I$, the quotient vector space $\mathfrak{g}/I$ has Lie bracket $[x+I,y+I]=[x,y]+I$. The ideal condition makes this independent of representatives. Its Lie algebra ideals correspond to the ideals of $\mathfrak{g}$ containing $I$, by inverse image under the quotient Lie algebra homomorphism.