Question on complexification of $\mathfrak sl(2,\Bbb C)$ As six generators of the real Lie algebra $\mathfrak sl(2,\Bbb C)_\Bbb R$ I can use the Pauli matrixes as follow:
$X_1=\frac{1}{2} \sigma_1, X_2=\frac{1}{2} \sigma_2, X_3=\frac{1}{2} \sigma_3$
$Y_1=\frac{1}{2}i \sigma_1, Y_2=\frac{1}{2}i \sigma_2, Y_3=\frac{1}{2}i \sigma_3$
cause it's easy to see that they have null traces and they are linearly independent on $\Bbb R$ span.
Based on Hall "Lie Groups, Lie Algebras, and Representations" page 66, a real Lie algebra $\mathfrak g $ of complex $n \times n $ matrix can be complexified only if $iX \notin \mathfrak g $ for every $X \in \mathfrak g$.
The generators above obviously doesn't satisfy that conditions, cause $iX_i = Y_i \in \mathfrak sl(2,\Bbb C)_\Bbb R$.
So it seems I can't complexify $\mathfrak sl(2,\Bbb C)_\Bbb R$ using the generators above.
However, on some books of QFT the complexification of $\mathfrak sl(2,\Bbb C)_\Bbb R$ is mentioned using as generators the standard generators of the rotations and boosts, and it's mentioned also in the following Wiki:
https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group
So my questions are:

*

*Does the complexification exist or doesn't exist depending on the generators used?

*Is $\mathfrak sl(2,\Bbb C)_\Bbb C \cong sl(2,\Bbb C)_\Bbb R \oplus sl(2,\Bbb C)_\Bbb R$ ?

 A: As regards question 2, you seem to be using subscripts for two different things. If subscript $\mathbb C$ stands for complexification but subscript $\mathbb R$ stands for scalar restriction to $\mathbb R$, then a) please use a different notation and b) it is still not true. Cf. answers to Are Lie algebra complexifications $\mathfrak g_{\mathbb C}$ equivalent to Lie algebra structures on $\mathfrak g\oplus \mathfrak g$? and Is the complexification of $\mathfrak{sl}(n, \mathbb{C})$ itself?.
What is true is
$$sl(2, \mathbb C) \otimes_{\mathbb R} \mathbb C\simeq sl(2, \mathbb C) \oplus sl(2, \mathbb C)$$
where both sides are complex Lie algebras and the isomorphism is one of complex LAs. If you view both sides as real Lie algebras via scalar restriction (and we denote that by the subscript $\mathbb R$), then this implies, of course, that there is an isomorphism of real Lie algebras
$$(sl(2, \mathbb C) \otimes_{\mathbb R} \mathbb C)_\mathbb R \simeq sl(2, \mathbb C)_{\mathbb R} \oplus sl(2, \mathbb C)_{\mathbb R}.$$
[So the left hand side is the complexification but viewed as a real algebra again, i.e. complexification followed by scalar restriction; if you insist on sticking to your confusing notation -- which you shouldn't -- you should write the left hand side as $(sl(2, \mathbb C)_{\mathbb C})_{\mathbb R}$.]
