Well I feel like this type of question has been asked over and over again, but I can't find any consistent answer in books or on this forum.
I'm studying the Lorentz group and its representations for the purposes of doing QFT calculations. I've run into massive difficulty in trying to connect the basic formalism that mathematicians have developed, and the calculations that are done by physicists.
The 3 resources that I've been sticking to are Peter Woits book on quantum theory and groups, this set of lecture notes: https://www.damtp.cam.ac.uk/user/examples/3P2.pdf and the Wikipedia article on the representation theory of the Lorentz group https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group#Conventions_and_Lie_algebra_bases.
On page 63 of the Cambridge paper the author remarks that Lie algebra of the Lorentz group is the same (I assume isomorphic) to $Re(\mathfrak{su}(2)\otimes\mathbf{C})$. He then goes on to complexify once again $Re(\mathfrak{su}(2)\otimes\mathbf{C})\otimes \mathbf{C}$, by defining $Z_{i}=\frac{1}{2}(J_{i}-iK_{i})$ and its conjugate. Each of the Z's satisfies the algebra of $\mathfrak{su}(2)$. When looking at the wiki article, they do this as well, complexifying the $\mathfrak{so}(1,3)$, and indicate that each of the raising and lowering operators obey the algebra of $\mathfrak{su}(2)$. Why is it then that they complexify again? The author of the wiki article explicitly says "Let $A_{C}$ and $B_{C}$ denote the complex linear span of A and B respectively." They then write $\mathfrak{so}(1,3)\otimes \mathbf{C}\cong (\mathfrak{su}(2)\otimes\mathbf{C})\oplus (\mathfrak{su}(2)\otimes\mathbf{C})$. Does this mean they are further defining raising and lowering operators using $A$ and $B$? Like $N=A \pm iB$?
I would like an explanation, using physics conventions (as found in Peskin and Schroeder) as to what precisely I'm calculating in QFT. In particular, I'd like to know, starting from the familiar representations of $\mathfrak{su}(2)$ that I know from quantum mechanics (in the physicist's convention), how I can obtain representations of the complexification of the Lorentz group. I'm not interested in rigorous proofs, I just want to know what in the world I'm actually calculating when I compute these quantities $exp(-\frac{i}{2}\omega_{\mu\nu}J^{\mu\nu})$ or $exp(-\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu})$ and how this relates to the various isomorphisms given in the wiki article. Please give a simple example, like in the $(\frac{1}{2},0)$ representation, the Schwartz QFT book says that $B_{i}=\frac{1}{2}\sigma_{i}$ and $A_{i}=0$ and then inverts the relation on the $A$ and $B$ to obtain $J_{i}=\frac{1}{2}\sigma_{i}$ and $K_{i}=\frac{i}{2}\sigma_{i}$. Please explain this in some detail if possible.