Let $G$ be a compact, simply-connected, and simple Lie group. Let $H$ be a closed subgroup of $G$ that has the same centralizer as the center of $G$. Is there a nice classification of such subgroups?

Edit: Given Misha's example, it seems that classifying such subgroups for all simple groups is hard. So let me pose the following questions:

1- Are there a maximal torus $T$ and $h \in H$ such that $h\in T$ and $h$ is not fixed by any non-trivial element of the Weyl group corresponding to the chosen torus?

2- Is there $h\in H$ such that $h$ is a general element of $G$?

Of course, 2 is a special case of 1.

Let $G$ be a compact, simply-connected, and simple Lie group. Let $H$ be a closed subgroup of $G$ that has the same centralizer as the center of $G$. Is there a nice classification of such subgroups?

Edit: Given Misha's example, it seems that classifying such subgroups for all simple groups is hard. So let me pose the following question:

Are there a maximal torus $T$ and $h \in H$ such that $h\in T$ and $h$ is not fixed by any non-trivial element of the Weyl group corresponding to the chosen torus?

Let $G$ be a compact, simply-connected, and simple Lie group. Is it possible that $G$ admits a closed proper subgroup which has the same centralizer as the center of $G$. If yes, is there a nice classification of such subgroups?

Let $G$ be a compact, simply-connected, and simple Lie group. Let $H$ be a closed subgroup of $G$ that has the same centralizer as the center of $G$. Is there a nice classification of such subgroups?

Edit: Given Misha's example, it seems that classifying such subgroups for all simple groups is hard. So let me pose the following questions:

1- Are there a maximal torus $T$ and $h \in H$ such that $h\in T$ and $h$ is not fixed by any non-trivial element of the Weyl group corresponding to the chosen torus?

2- Is there $h\in H$ such that $h$ is a general element of $G$?

Of course, 2 is a special case of 1.