How do I construct the $SU(2)$ representation of the Lorentz Group using $SU(2)\times SU(2)\sim SO(3,1)$? This question is based on  problem II.3.1 in Anthony Zee's book Quantum Field Theory in a Nutshell

Show, by explicit calculation, that $(1/2,1/2)$ is the Lorentz Vector.

I see that the generators of $SU(2)$ are the Pauli matrices and the generators of $SO(3,1)$ is a matrix composed of two Pauli matrices along the diagonal. Is it always the case that the direct product of two groups is formed from the generators like this?
I ask this because I'm trying to write a Lorentz boost as two simultaneous quaternion rotations [unit quaternions rotations are isomorphic to $SU(2)$] and tranform between the two methods. Is this possible?
In other words, How do I construct the $SU(2)$ representation of the Lorentz Group using the fact that $SU(2)\times SU(2) \sim SO(3,1)$?
Here is some background information:
Zee has shown that the algebra of the Lorentz group is formed from two separate $SU(2)$ algebras [$SO(3,1)$ is isomorphic to $SU(2)\times SU(2)$] because the Lorentz algebra satisfies:
$$\begin{align}[J_{+i},J_{+j}] &= ie_{ijk}J_{k+} &
[J_{-i},J_{-j}] &= ie_{ijk} J_{k-} &
[J_{+i},J_{-j}] &= 0\end{align}$$
The representations of $SU(2)$ are labeled by $j=0,\frac{1}{2},1,\ldots$ so the $SU(2)\times SU(2)$ rep is labelled by $(j_+,j_-)$ with the $(1/2,1/2)$ being the Lorentz 4-vector because and each representation contains $(2j+1)$ elements so $(1/2,1/2)$ contains 4 elements.
 A: For the problem at hand formulated in a precise manner, „Show that the $\left(\frac{1}{2},\frac{1}{2}\right)$ representation of the $\mbox{SL}(2,\mathbb{C})$ group is* the Lorentz 4-vector", the solution - which is not so apparent from Qmechanic's otherwise good post - should be exhibited by direct / brute-force computation. This is relatively easy, and I quote from my diploma/Batchlor's Degree graduation paper (written in my native Romanian)
PART 1:
Let $\psi_{\alpha}$ be the components of a Weyl spinor wrt the canonical basis in a 2-dimensional vector space in which the fundamental $\left(\frac{1}{2},0\right)$ representation of $\mbox{SL}(2,\mathbb{C})$ "lives". Idem for $\bar{\chi}_{\dot{\alpha}}$ and the contragradient representation of the same group, $\left(0,\frac{1}{2}\right)$. Then, as an application of the Clebsch-Gordan theorem for $\mbox{SL}(2,\mathbb{C})$:
LEMMA: 
$\begin{equation}
\psi _{\alpha }\otimes \overline{\chi }_{\stackrel{\bullet }{\alpha }}\equiv
\psi _{\alpha }\overline{\chi }_{\stackrel{\bullet }{\alpha }}=\left[ \frac{1%
}{2}\psi ^{\beta }\left( \sigma ^{\mu }\right) _{\beta \stackrel{\bullet }{%
\beta }}\overline{\chi }^{\stackrel{\bullet }{\beta }}\right] \left( \sigma
_{\mu }\right) _{\alpha \stackrel{\bullet }{\alpha }}\equiv V^{\mu}\left( \sigma
_{\mu }\right) _{\alpha \stackrel{\bullet }{\alpha }}\text{.}  
\end{equation}$
PROOF:
$\left[ \frac{1}{2}\psi ^{\beta }\left( \sigma _{\mu }\right) _{\beta 
\stackrel{\bullet }{\beta }}\overline{\chi }^{\stackrel{\bullet }{\beta }%
}\right] \left( \sigma ^{\mu }\right) _{\alpha \stackrel{\bullet }{\alpha }}=%
\frac{1}{2}\left( \varepsilon ^{\beta \gamma }\psi _{\gamma }\right) \left(
\sigma ^{\mu }\right) _{\beta \stackrel{\bullet }{\beta }}\left( \varepsilon
^{\stackrel{\bullet }{\beta }\stackrel{\bullet }{\gamma }}\overline{\chi }_{%
\stackrel{\bullet }{\gamma }}\right) \left( \sigma _{\mu }\right) _{\alpha 
\stackrel{\bullet }{\alpha }} \\
=-\frac{1}{2}\psi _{\gamma }\varepsilon ^{\beta \gamma }\varepsilon ^{%
\stackrel{\bullet }{\gamma }\stackrel{\bullet }{\beta }}\left( \sigma ^{\mu
}\right) _{\beta \stackrel{\bullet }{\beta }}\overline{\chi }_{\stackrel{%
\bullet }{\gamma }}\left( \sigma _{\mu }\right) _{\alpha \stackrel{\bullet }{%
\alpha }} \\
=\frac{1}{2}\psi _{\gamma }\left[ \varepsilon ^{\gamma \beta }\varepsilon ^{%
\stackrel{\bullet }{\gamma }\stackrel{\bullet }{\beta }}\left( \sigma ^{\mu
}\right) _{\beta \stackrel{\bullet }{\beta }}\right] \overline{\chi }_{%
\stackrel{\bullet }{\gamma }}\left( \sigma _{\mu }\right) _{\alpha \stackrel{%
\bullet }{\alpha }} \\
=\frac{1}{2}\psi _{\gamma }\overline{\chi }_{\stackrel{\bullet }{\gamma }%
}\left( \overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\gamma }%
\gamma }\left( \sigma _{\mu }\right) _{\alpha \stackrel{\bullet }{\alpha }}
\\
=\psi _{\gamma }\overline{\chi }_{\stackrel{\bullet }{\gamma }}\delta _{%
\stackrel{\bullet }{\alpha }}^{\stackrel{\bullet }{\gamma }}\delta _{\alpha
}^{\gamma }=\psi _{\alpha }\overline{\chi }_{\stackrel{\bullet }{\alpha }}
$
This proof makes the Pauli matrices to be seen as Clebsch-Gordan coefficients. 
PART 2: 
THEOREM:
$V^{\mu}\left(\psi,\chi\right)$ defined above is a Lorentz 4-vector (i.e. they are components of a Lorentz 4-vector seen as a generic member of a vector space carrying the fundamental representation of the restricted Lorentz group $\mathfrak{Lor}(1,3)$). 
PROOF:
$V'^{\mu}\equiv 
\left( \phi ^{\prime }\right) ^{\alpha }\left( \sigma ^{\mu }\right)
_{\alpha \stackrel{\bullet }{\beta }}\left( \overline{\chi }^{\prime
}\right) ^{\stackrel{\bullet }{\beta }}=-\left( \overline{\chi }^{\prime
}\right) _{\stackrel{\bullet }{\alpha }}\left( \overline{\sigma }^{\mu
}\right) ^{\stackrel{\bullet }{\alpha }\beta }\left( \phi ^{\prime }\right)
_{\beta }=-\left( M^{*}\right) _{\stackrel{\bullet }{\alpha }}{}^{\stackrel{%
\bullet }{\beta }}\overline{\chi }_{\stackrel{\bullet }{\beta }}\left( 
\overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\alpha }\beta
}M_{\beta }{}^{\gamma }\phi _{\gamma } \\
=-\overline{\chi }_{\stackrel{\bullet }{\beta }}\left( M^{\dagger }\right) ^{%
\stackrel{\bullet }{\beta }}{}_{\stackrel{\bullet }{\alpha }}\left( 
\overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\alpha }\beta
}M_{\beta }{}^{\gamma }\phi _{\gamma } \\
=-\overline{\chi }_{\stackrel{\bullet }{\beta }}\delta _{\stackrel{\bullet }{%
\gamma }}^{\stackrel{\bullet }{\beta }}\left( M^{\dagger }\right) ^{%
\stackrel{\bullet }{\gamma }}{}_{\stackrel{\bullet }{\alpha }}\left( 
\overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\alpha }\beta
}M_{\beta }{}^{\gamma }\delta _{\gamma }^{\zeta }\phi _{\zeta } \\
=-\frac{1}{2}\overline{\chi }_{\stackrel{\bullet }{\beta }}\left( \overline{%
\sigma }^{\nu }\right) ^{\stackrel{\bullet }{\beta }\zeta }\left( \sigma
_{\nu }\right) _{\gamma \stackrel{\bullet }{\gamma }}\left( M^{\dagger
}\right) ^{\stackrel{\bullet }{\gamma }}{}_{\stackrel{\bullet }{\alpha }%
}\left( \overline{\sigma }^{\mu }\right) ^{\stackrel{\bullet }{\alpha }\beta
}M_{\beta }{}^{\gamma }\phi _{\zeta } \\
=-\frac{1}{2}\left[ \left( M^{\dagger }\right) ^{\stackrel{\bullet }{\gamma }%
}{}_{\stackrel{\bullet }{\alpha }}\left( \overline{\sigma }^{\mu }\right) ^{%
\stackrel{\bullet }{\alpha }\beta }M_{\beta }{}^{\gamma }\left( \sigma _{\nu
}\right) _{\gamma \stackrel{\bullet }{\gamma }}\right] \left[ \overline{\chi 
}_{\stackrel{\bullet }{\beta }}\left( \overline{\sigma }^{\nu }\right) ^{%
\stackrel{\bullet }{\beta }\zeta }\phi _{\zeta }\right] \\
=-\frac{1}{2}Tr\left( M^{\dagger }\overline{\sigma }^{\mu }M\sigma _{\nu
}\right) \left( \overline{\chi }\overline{\sigma }^{\nu }\phi \right) \\
=-\Lambda ^{\mu }{}_{\nu }\left( M\right) \left( \overline{\chi }\overline{%
\sigma }^{\nu }\phi \right) \\
=\Lambda ^{\mu }{}_{\nu }\left( M\right) \left( \phi \sigma ^{\nu }\overline{%
\chi }\right) \equiv \Lambda ^{\mu }{}_{\nu }\left( M\right) V^{\nu} $
*is = in the sense of group representation theory, it means that the carrier vector spaces of the two representations are isomorphic which is the content of the lemma. 
Note to the reader: the proof of the theorem uses the fact that these "classical" spinors have Grassmann parity 1. This explains the appearance and disappearance of the "-" sign. 
