Center of Lie algebra and Lie algebra of center

Question

I am learning some Lie group and I met the following problem:

Let $G$ be a Lie group and $Z$ its center. Then $Z$ is also a Lie group. Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\mathfrak{z}$ be the center of $\mathfrak{g}$. Is the Lie algebra of $Z$ equal to $\mathfrak{z}$?

It seems that this correspondence is natural, yet I cannot find any source containing this. Could anyone suggest a (sketch of) proof, or give a counter example?

Thank you very much!

Answer 1

Among nonconnected Lie groups, $O(2)$ is a counter example. For connected Lie groups the equality holds, see the link given by Burde.