Centralizer of an element in a matrix Lie group whose Jordan form is given $\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}$Let $G\subset \GL_n(\mathbb{C})$ be a complex matrix Lie group, (e.g., $\SO_n$, $\Sp_{n}$), and let $x\in G$ with Jordan form $u$ ($u$ may not belong to $G$) given, I wonder how to compute the centralizer $Z_{G}(x)$.
I know the Lie algebra of $Z_{G}(x)$ is $\{Y\in \mathfrak{g}\mid \Ad(x)Y=Y\}$. But it's hard for me to find a matrix $t\in \GL_n(\mathbb{C})$ such that $tut^{-1}\in G$. I tried to find $U\in \mathfrak{gl}_n(\mathbb{C})$ such that $\exp(U)=u$ and then find a $T\in \mathfrak{gl}_n(\mathbb{C})$ such that $\ad(T)(U)\in \mathfrak{g}$, but I don't think that means $\exp(T)u(\exp(T))^{-1}\in G$.
Any help will be appreciated.
 A: You’re question talks abut the centraliser but you then seem instead to be unsure how to relate an arbitrary Jordan normal to an element of $G$. I’ll try to answer both.
Firstly, the centraliser. A matrix of the form:
$$ \begin{pmatrix}\lambda & 1 & & \\
& \ddots &\ddots \\ & & \ddots &1 \\ & & & \lambda \end{pmatrix}$$ has centraliser given by upper triangular Toeplitz matrices e.g. :
$$ \begin{pmatrix}a & b & c & d\\
& a &b & c \\ & & a &b \\ & & & a \end{pmatrix} $$
So the centraliser of a matrix in Jordan normal form comprises block diagonal matrices with these Toeplitz blocks where the generalised eigenspaces are and generic blocks where the honest eigenspaces are.
Now if we have a $v$ conjugate to $u$, their centralisers are conjugate as well by the same element so we just need fo find a way to relate $u$ to an element of $G$. In practice this is a bit tedious so I'll be brief. We just need to pick $v$ with the right eigenvalues, dimension of eigenspaces and generalised eigenspaces, etc. We then find a basis of eigenvectors and generalised eigenvectors (arranged in Jordan chains) and then our conjugacy matrix has these as columns.
Of course there will be many such $v$ in general so it would usually make more sense to start with $v$ find its Jordan normal form and use that to compute the centraliser.
