Recently I have been studying Lie algebra, and I have been solving some problems to get familiarized with the concepts.
A problem that I recently solved was to prove that in a global sense, the exponential map: $exp:\mathfrak{sl}(2,\mathbb{C})\rightarrow SL(2,\mathbb{C})$ is not a diffeomorphism (which is not complicated as it is sufficient to prove that the exponential is not surjective in a point).
After that me profesor suggested me to prove the following:
Find an element $x\in\mathfrak{sl}(2,\mathbb{C})$ sucht that the exponential is not a local diffeomorphism at $x$.
For this, I got a hint from the profesor, which was to find a point where the kernel of the differential of the exponential is non trivial. I think that this could help me prove that the exponential is not injective. ( I'm not sure)
Here is the definition of the differential of exponential that I found on the internet $$\frac{d}{dt}e^{X(t)}=e^{X(t)}\frac{1-e^{1-ad_{X}}}{ad_{X}}\frac{dX(t)}{dt}.$$
The problem is that this is the first time I try to use this formula, and I am unsure of how to proceed.
So, I would appreciate any help, also diferent ideas are welcomed.