Short Exact Sequence and Solvable Lie Groups.

Solution 1:

If $\mathfrak h \rightarrow \mathfrak g$ is the usual inclusion map, and $\mathfrak g \rightarrow \mathfrak g/ \mathfrak h$ is the usual projection map, then

$$0 \rightarrow \mathfrak h \rightarrow \mathfrak g \rightarrow \mathfrak g / \mathfrak h \rightarrow 0$$

is an exact sequence. If you think about what it means for a sequence of lie algebras (or more generally, of vector spaces) to be exact, then this is basically what all short exact sequences look like: the first term is a subspace of the second term, and the third term is the quotient of the second term by the first.