For any regular element $g$ in $\mathfrak{gl}_n(C)$, the dimension of centralizer of $g$ will be $n$. What's more, the centralizer of regular elements can be simultaneously generated by homogeneous invariant polynomial of $\mathfrak{gl}_n(C)$, hence it's isomorphic to $\exp{(\mathbb{C}^n)}$. For a reference, see lemma $6$ of paper:https://link.springer.com/content/pdf/10.1023/A:1006567614081.pdf
Notice that regularity can be characterized by degree of minimal polynomial. Then for a fixed natural number $d$, let's consider a subspace $S_d$ of $\mathfrak{gl}_n(C)$ consisting of element in $\mathfrak{gl}_n(C)$ whose degree of minimal polynomial is given by $d$. Is the dimension of centralizer also a fixed number? Can we find a Lie group so that it generates centralizer of $S_d$ simultaneously?
Also, I wonder do we have similar result for all complex Lie algebra, in cluding $\mathfrak{gl}_n(C)$ we consider above.