Intersection of a family of closed Lie subgroups

Question

If I have a Lie group $G$, and $\{H_{\alpha}\}_{\alpha\in A}$ is a family of closed Lie subgroups with Lie algebra $\{\mathfrak{h}_{\alpha}\}_{\alpha\in A}$, it's easy to see that $\bigcap_{\alpha} H_{\alpha}$ is a Lie subgroup since it's a closed subgroup, but what about it's Lie algebra $\mathfrak{h}$? Is it always true to say that $\mathfrak{h}=\bigcap_{\alpha} \mathfrak{h_{\alpha}}$? ( $\mathfrak{h}\subset\bigcap_{\alpha} \mathfrak{h_{\alpha}}$ is easy, but what about the other direction?)If it's not true, any counterexample?