Can we always choose a complementary Lie algebra such that the associated group is closed?

Question

Let $G$ be a compact Lie group and $H$ a closed normal subgroup of codimension 1. As $H$ is normal it's associated Lie algebra $\mathfrak{h}$ is an ideal of $\mathfrak{g}$ and one can find a complementary ideal $\mathfrak{k}$ such that $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{k}$. It is clear that sometimes this choice can go "wrong" and result in $\exp(t\mathfrak{k})$ being a non-closed subgroup (e.g. quasi periodic winding on $\mathbb{T}^2$), but I feel like there should always be a "right" choice leading to a closed subgroup. Is this correct?