Suppose I have a Lie group $G$ and a closed normal subgroup $H$, both connected. Then I can form the quotient $G/H$, which is again a Lie group. On the other hand, the derivative of the embedding $H\hookrightarrow G$ gives an embedding of Lie algebras, $\mathfrak{h}\hookrightarrow\mathfrak{g}$. Since $H$ is normal, $\mathfrak{h}$ is an ideal of $\mathfrak{g}$, so we can form the quotient algebra $\mathfrak{g}/\mathfrak{h}$.

Is it then true that the Lie algebra of $G/H$ is canonically isomorphic to $\mathfrak{g}/\mathfrak{h}$? (I guess what I really want to know is: is the functor $\phi\mapsto d_e\phi$ exact?)

If so, can we always write $\mathfrak{g}$ as a direct sum of vector spaces $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{g}'$ where $\mathfrak{g}'$ is a subalgebra of $\mathfrak{g}$ isomorphic to $\mathfrak{g}/\mathfrak{h}$? (I don't mean this to be a direct sum of Lie algebras; there would certainly in general be a nontrivial bracket between $\mathfrak{g}'$ and $\mathfrak{h}$.)

Thank you.