Using answer to my previous question I made some progress towards understanding Lie algebra homomorphisms. But of course I am unsure whether my thoughts are really correct so again I'd like to request the community to check my thoughts.
Is the following correct?
It is my goal to find a Lie algebra isomorphism from the Lie algebra of $SL_2(\mathbb C)$ to the Lie algebra of $O(3, \mathbb C)$.
The group $G = SL_2(\mathbb C)$ is both connected and simply connected. The following is a theorem (it can be found e.g. here):
For Lie groups $G, H$ with $G$ connected and simply connected, a linear map $\varphi : \mathfrak g \to \mathfrak h$ is the derivative of a homomorphism $\phi : G \to H$ if and only if φ is a Lie algebra homomorphism.
Hence if $\phi: SL_2 \to O(3,\mathbb C)$ is a Lie group homomorphism, its derivative will yield a Lie group homomorphism.
(If $\phi: SL_2 \to O(3,\mathbb C)$ is a Lie group isomorphism is its derivative a Lie algebra isomorphism?)
Therefore, to find the desired Lie algerba isomorphism I pick
a basis generators for $SL_2$ and a basis generators for $O(3, \mathbb C)$, define a map on the generators and then take its derivative. But the derivative of a linear map is the linear map itself here. This leads me to believe that I made a mistake somewhere. Or did I not?
In response to Ben's comment: Let's replace "basis" with "generators" instead.