Given a short exact sequence of groups
$ 0 \to B \to G \to G/B \to 0$, such that $G = B \oplus G/B$,
does there exists a corresponding short exact sequence of lie algebras
$0 \to \mathfrak{b} \to \mathfrak {g} \to \mathfrak{g/b} \to 0$ such that $\mathfrak{g} = \mathfrak{b} \oplus \mathfrak{g/b}$?
For simplicity we can assume $G$ is $SL_n$, $B$ is a Borel subgroup and $G/B$ is the quotient.