TITLE: Centralizer of one element on a compact connected Lie group
QUESTION [10 upvotes]: Exercise 16.2 from Daniel Bump - Lie Groups.
Let $G$ be a compact connected Lie group and let $g\in G$. Show that
the centralizer $C_{G(g)}$ of $g$ is connected.
I have some problems verifying this, i have tried to use that in that case the exponential map is surjective, and things of maximal torus, but i have not achieved it, i would appreciate some answer.

REPLY [3 votes]: This is not true in general. Consider $SO(3)$ and the matrix $$ \begin{bmatrix} 
  1 & 0 & 0\\ 0 & -1 & 0 \\ 0&0&-1
\end{bmatrix} $$
Then $C_G(g)=\left\{ \begin{bmatrix} 
  a & 0 & 0\\ 0 & \pm \cos \theta & \mp \sin \theta \\ 0&\sin\theta&\cos\theta
\end{bmatrix}: a\in \{\pm 1\}, \theta\in \mathbb{R}  \right\}   $.
We can fix the problem statement by adding that $G$ is simply connected. Then we can argue using a long exact sequence in homotopy groups.