The following is stated as Theorem 3.1(i) on p. 50f. in Onishchik/Vinberg, Lie groups and Lie algebras III:
Let $G$ be a connected real Lie groups. If there is a closed connected normal subgroup $G_1$ of codimension $1$ , then there exists a one-dimensional Lie subgroup $C$ of $G$ such that $G$ is the semidirect product of $C$ and $G_1$.
It is easy to see that if $G$ is abelian, then the statement is true. It is also easy to see that $G_1$ needs to contain the commutator subgroup of $G$ and hence its closure.
Onishchik/Vinberg now claim that one can reduce the general statement to the abelian case by induction on the dimension of $G$. I don't see how this is supposed to work. Any hints?