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?
