TITLE: Precise connection between complexification of $\mathfrak{su}(2)$, $\mathfrak{so}(1,3)$ and $\mathfrak{sl}(2, \mathbb{C})$
QUESTION [19 upvotes]: I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by
\begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation}
\begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation}
\begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation}
We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy
\begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation}
\begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation}
\begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation}
where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:

When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation}
We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$:
$(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here?
$\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$:
\begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation}
Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$
Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$
Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?

I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$. Wikipedia  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great.

REPLY [6 votes]: I came up with the same question and this link helped me a lot https://en.wikiversity.org/wiki/Representation_theory_of_the_Lorentz_group#The_Lie_algebra. Look at the isomorphism chain in (A1) and read those passages. Also, Brian C. Hall's book "Lie groups, lie algebras, and representations." helps.
I think the confusion comes from not distinguishing between complex-linear (C-linear) and real-linear (R-linear) representations of the algebras. Every representation I will be talking about below is of finite dimension, V is a complex vector space.

R-linear representation of [the real lie algebra] su(2) over V has 1-1 correspondence with C-linear representation of complexification of [the real lie algebra] su(2). 

i.e. R-linear rep. of [real lie algebra] su(2) over V has 1-1 correspondence with C-linear rep. [complex lie algebra] sl(2,C) over V.

R-linear rep. of [real lie algebra] so(1,3) over V has 1-1 correspondence with C-linear rep. of complexification of [real lie algebra] so(1,3) over V.

Complexification of [real lie algebra] so(1,3) is isomorphic to the direct sum of two copies of [complex lie algebra] sl(2,C), which in turn is isomorphic to the complexification of [complex lie algebra] sl(2,C). 

C-linear rep. of complexification of [complex lie algebra] sl(2,C) over V has 1-1 correspondence with R-linear rep. of  the decomplexification of [complex lie algebra] sl(2,C) over V. 

Hence, R-linear rep. of [real Lie algebra] so(1,3) over V has 1-1 correspondence with R-linear rep. of the decomplexification of [complex lie algebra] sl(2,C) over V. 
For me this makes sense, but if anyone could verify it as well it would be great.