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I am studying the orthochronous Lorentz algebra $\mathfrak{so}(3,1)^\uparrow $ and it reads

$$ [X_i,X_j]=i \varepsilon_{ijk} X_k $$ $$ [X_i,Y_j]=i \varepsilon_{ijk} Y_k $$ $$ [Y_i,Y_j]=-i\varepsilon_{ijk}X_k $$

If I complexify this by taking complex combinations of the form

$$ X_i^\pm = \frac{1}{2}(X_i \pm iY_i)$$

I find that I have two $\mathfrak{su}(2)$ Lie algebras:

$$ [X^+_i,X^+_j]=i \varepsilon_{ijk} X^+_k $$ $$ [X^-_i,X^-_j]=i \varepsilon_{ijk} X^-_k $$ $$ [X^+_i,X^-_j]=0 $$

which is two $\mathfrak{su}(2)$ subalgebras, so I should be able to say that $\mathfrak{so}(3,1)_{\mathbb{C}}^\uparrow \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$. However I have read in a lot of places, including here, that $\mathfrak{so}(3,1)_{\mathbb{C}}^\uparrow \cong \mathfrak{su}_\mathbb{C}(2) \oplus \mathfrak{su}_\mathbb{C}(2)$. Why do the $\mathfrak{su}(2)$s have to be complexified? The commutators of $\{ X_i^\pm, Y_j^\pm \} $ are $\mathfrak{su}(2)$, why can I complexify them? and why should they be? Is my decomposition not correct?

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    $\begingroup$ For starters, check the dimensions: The complex Lorentz algebra $\mathfrak{so}(3,1)_{\mathbb{C}}$ has 6 complex dimensions = 12 real dimensions, while $\mathfrak{su}(2)$ has only 3 real dimensions, so a conjectured isomorphism $\mathfrak{so}(3,1)_{\mathbb{C}} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ cannot possibly be true. $\endgroup$ – Qmechanic Jun 8 '18 at 18:01
  • $\begingroup$ Essentially a duplicate of physics.stackexchange.com/q/28505/2451. $\endgroup$ – Qmechanic Jun 8 '18 at 18:11
  • $\begingroup$ @Qmechanic Ah that makes sense, thank you. If they weren't the complexified version too, I would not be able to label my representations by the highest weight because that only applies for complexified Lie algebras doesn't it? $\endgroup$ – Matt0410 Jun 8 '18 at 18:18
  • $\begingroup$ $\uparrow$ Right. $\endgroup$ – Qmechanic Jun 8 '18 at 18:19