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### Show the Lie algebra is the same for $SU(2) \times SU(2)$ and Lorentz group [duplicate]

So I know: $$[\sigma_{I},\sigma{j}] = 2i \epsilon_{ijk} \sigma_{k}$$ So two products of this should give us the Lorentz group: $SO(4) = SU(2) \times SU(2)$ Where $SO(4)$ has 3 Lie algebra which can ...
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### Why study $SL(2,\mathbb{C})$ representations when the representations of $SU(2)\times SU(2)$ exhausts all irreps of $SO(3,1)$? [duplicate]

One can show that the Lie algebra of $SO(3,1)$ is isomorphic to $SU(2)\times SU(2)$. And the representations of $SU(2)\times SU(2)$ exhausts all possible representations of $SO(3,1)$. $\bullet$ Why ...
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### Why do we use the complexification of the Lorentz group?

I do understand why we are using the double cover, but why exactly do we make the transition to complex Lorentz transformations? Where and why are they needed? To be precise: The double cover of ...
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### Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$?

The number of generators of Lie algebra $sl(2,\mathbb{C})$ is 6, and $sl(2,\mathbb{R})$ has 3 generators, Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$? Say \...
The quaternion Lorentz boost $v'=hvh^*+ 1/2( (hhv)^*-(h^*h^*v)^*)$ where $h$ is $(\cosh(x),\sinh(x),0,0)$ was derived by substituting the hyperbolic sine and cosine for the sine and cosine in the ...