There have been a lot of questions in the past about the subtleties of the Lorentz algebra. In particular the usage of the real Lie algebra $\mathfrak{so}(1,3;\Bbb R)$ and it's complexification which is more convenient in the context of representation theory. I have searched through a lot of questions on the site and I would like to just outline a few things I've picked up and ask that if there is anything incorrect in the following, could someone please point it out.
The following Lie algebra isomorphisms are useful:
$$\mathfrak{so}(1,3;\Bbb R)_{\Bbb C}\cong\mathfrak{so}(1,3;\Bbb C)\cong \mathfrak{sl}(2;\Bbb C)\oplus \mathfrak{sl}(2;\Bbb C)\cong \mathfrak{su}(2)_{\Bbb C}\oplus \mathfrak{su}(2)_{\Bbb C}\tag{1}.$$
In particular the first with the last, as the representation theory of $\mathfrak{su}(2)_{\Bbb C}$ is well understood, and hence the classifications of the representations of the (complexified) Lorentz algebra can be obtained.
Next I note two forms that I have seen of the Lie bracket relations for the Lorentz algebra:
$[J_i,J_j]=i\epsilon_{ijk}J_k, \quad [K_i,K_j]=-i\epsilon_{ijk}J_k,\quad [J_i,K_j]=i\epsilon_{ijk}K_k \tag{2}$
$[J_i,J_j]=\epsilon_{ijk}J_k, \quad [K_i,K_j]=-\epsilon_{ijk}J_k,\quad [J_i,K_j]=\epsilon_{ijk}K_k \tag{3}$
Found here on page 3 and here on page 22 respectively. These are obviously very similar and only differ by a factor of $i$. My question is (since the subtlety of whether we are working with the Lorentz algebra or the complexified Lorentz algebra is almost never said in physics texts), is it sufficient in passing from the real Lorentz algebra to the complexified Lorentz algebra, to simply insert a factor of $i$ into the Lie bracket relations, as above? Or is there something more subtle being done here?
The text in which the first set of relations is found, still defines the following quantities:
$$J_j^{\pm}:=\frac{1}{2}(J_j\pm iK_j) \tag{4},$$
which is surprising to me since it appears we are already in the complexified Lorentz algebra, given the commutation relations they give in $(2)$. In other words, why is it necessary to take a complex linear combination of $J$ and $K$ here just as we would if we were going from the real algebra to the complex. I understand that in terms of these new $J_j^\pm$ generators we find the isomorphism $\mathfrak{so}(1,3)_{\Bbb C}\cong \mathfrak{su}(2)_{\Bbb C}\oplus \mathfrak{su}(2)_{\Bbb C}$ and things follow from there.
Are the $J_j$ and $K_j$ above perhaps not the same as in $(2)$?