Let $\mathfrak{g}$ and $\mathfrak{g'}$ be complex Lie algebras such that $\mathfrak{g}$ is a subalgebra of $\mathfrak{g'}$. Let $Z(\mathfrak{g})$ and $Z(\mathfrak{g'})$ be the centers of the universal enveloping algebras.

Is there any relation between $Z(\mathfrak{g})$ and $Z(\mathfrak{g'})$?

Is there any example where one has $Z(\mathfrak{g})\subseteq Z(\mathfrak{g'})$?