Let $\mathfrak{g}_1\subset \mathfrak{g}_2$ be a Lie algebra embedding. Assume both are semisimple. For instance take the standard diagonal embedding $\mathfrak{sl}(2, \mathbb{C})\subset \mathfrak{sl}(3, \mathbb{C})$. This lifts to an embedding $U(\mathfrak{sl}(2, \mathbb{C}))\subset U(\mathfrak{sl}(3, \mathbb{C}))$ of the corresponding enveloping algebras. Consider their centers $Z(\mathfrak{sl}(2, \mathbb{C})),Z(\mathfrak{sl}(3, \mathbb{C}))$. It is relatively easy to see that under the previous embedding we have that $Z(\mathfrak{sl}(2, \mathbb{C}))\cap Z(\mathfrak{sl}(3, \mathbb{C})) = \mathbb{C}$. Is there any reasonable way to "connect" these centers? Let me be a bit more clear. For instance, we can pass the same question to the symmetric algebras of $\mathfrak{g}_1,\mathfrak{g}_2$. Here we have $S(\mathfrak{sl}(2, \mathbb{C})^*)^{SL(2,\mathbb{C})}$ the algebra of $SL(2,\mathbb{C})$-invariant functions on $\mathfrak{sl}(2, \mathbb{C})$ and $S(\mathfrak{sl}(3, \mathbb{C})^{\*} )^{SL(3,\mathbb{C})}$. There is a natural embedding of $S(\mathfrak{sl}(3, \mathbb{C})^{\*} )^{SL(3,\mathbb{C})}$ into $S(\mathfrak{sl}(3, \mathbb{C})^{\*} )^{SL(2,\mathbb{C})}$ and a projection of $S(\mathfrak{sl}(3, \mathbb{C})^{\*} )^{SL(2,\mathbb{C})}$ onto $S(\mathfrak{sl}(2, \mathbb{C})^{\*} )^{SL(2,\mathbb{C})}$ given by restriction. Are there any results that could describe how these centers behave under restrictions?