Ideal of a Lie algebra

Definition with symbols
A subset $$B$$ of a Lie algebra $$A$$ is said to be an ideal if it is a vector subspace of $$A$$ under addition, and $$[x,y] \in B$$ for any $$x \in B$$ and $$y \in A$$. Note that any ideal is, in particular, also a Lie subalgebra.

Relation with the Lie group
The Lie algebra of any closed normal subgroup of a Lie group, is an ideal. This follows from the following:

The commutator of $$v$$ and $$w$$ is the differential at $$t = 0$$ of $$\exp(tv).w - w$$. Here $$\exp(tv)$$ acts on $$w$$ via the adjoint representation.

Now if $$w$$ is an element in the Lie algebra of a closed normal subgroup $$N$$, then $$\exp(tv)\exp(t'w)\exp(tv)^{-1}$$ is also in $$N$$ for every $$t'$$, and differentiating at $$t'=0$$ gives $$\exp{tv}.w$$ to be in the Lie algebra of $$N$$. Thus, the difference $$\exp(tv).w - w$$ is also in the Lie algebra, and hence so is the value $$[v,w]$$

Thus $$w$$ being in the Lie algebra of $$N$$ implies that $$[v,w]$$ is. By the amticommutativity of the Lie bracket, we conclude that $$v$$ being in the Lie algebra implies $$[v,w]$$ is.