I'm trying to prove the following: Suppose $G$ is a connected matrix Lie group with Lie algebra $\mathfrak{g}$ and $A$ is an element of $G$. Show that $A$ belongs to the center of $G$ if and only if $Ad_{A}(X) = X\;\forall\;X\in\mathfrak{g}$, where $Ad_{A}$ is the adjoint representation of $G$ on $\mathfrak{g}$.
This is what I've done so far:
Suppose $A$ is in the center of $G$. and let $X\in\mathfrak{g}$. Since $X\in\mathfrak{g}$, we know that $exp(tX)\in G; t\in\mathbb{R}$. Also, $exp(tX)$ is the unique one parameter subgroup generated by $X$ and so $X=\frac{d}{dt}(exp(tX))\mid_{t=0}$
$Ad_{A}(X) = AXA^{-1} = A\frac{d}{dt}exp(tX)\mid_{t=0}A^{-1} = \frac{d}{dt}Aexp(tX)A^{-1}\mid_{t=0}= \frac{d}{dt}exp(tX)AA^{-1}\mid_{t=0}= \frac{d}{dt}exp(tX)\mid_{t=0}= X$
The converse follows a similar thought process.
I want to know if this proof is correct. I'd also be glad if I could be pointed in the right direction if it's not correct. Any help is much appreciated