Let $Cl(1,3)$ be the Clifford algebra for $\mathbb{R}^{1,3}$, i.e. $\mathbb{R}^4$ equipped with a Minkowski scalar product of signature $(-,+,+,+)$. The complexification of the Clifford algebra is $\text{End}(\mathbb{C}^4)$, and there is thus a faithful representation of $Cl(1,3)$ on $\mathbb{C}^4$ given by the gamma block matrices:
\begin{align*} \Gamma_0=&\begin{pmatrix} 0&iI_2\\ iI_2&0 \end{pmatrix}\\ \Gamma_i=&\begin{pmatrix} 0&i\sigma_i\\ -i\sigma_i&0 \end{pmatrix}\text{ for all }i=1,2,3 \end{align*} here $\sigma_i$ are the Pauli spin matrices.
There is a chirality element given by: \begin{align*} \Gamma_5=&-i^3\Gamma_0\Gamma_1\Gamma_2\Gamma_3\\ =&\begin{pmatrix} -I_2&0\\ 0&I_2 \end{pmatrix} \end{align*} Which splits the space into a direct sum $\mathbb{C}^2\oplus\mathbb{C}^2$ corresponding to the eigenspaces of $\Gamma_5$. These spaces are called right (negative) and left (positive) handed Weyl spinors.
I am trying to figure out the representation of $SL_2(\mathbb{C})$ on $\mathbb{C}^2\oplus \mathbb{C}^2$ induced by the above faithful representation, but I am getting pretty confused.
First note that $SL_2(\mathbb{C})$ is the spin group $Spin^+(1,3)$, where $Spin^+(1,3)$ is the subgroup of invertible Clifford elements generated by elements of the form:
$$v_1\cdots v_{2q}$$
where an even number of $\langle v_i,v_i\rangle=1$ and even amount satisfy $\langle v_i,v_i\rangle=-1$. It follows that the induced representation $\kappa$ on any element of $Spin^+(1,3)$ is found by writing:
$$v_j= v_j^{i_j}e_{i_j}$$
where there is an implied sum over $i_j$ is implied, and then taking:
$$\kappa(v_1\cdots v_{2q})=v_{1}^{i_1}\cdots v_{2q}^{i_{2q}}\Gamma_{i_1}\cdots \Gamma_{i_{2q}}$$
In particular, $e_ie_j\in Spin^+(1,3)$ for $0<i\leq j\leq 3$. However, I calculate:
$$\Gamma_i\Gamma_j=\begin{pmatrix} \sigma_i\sigma_j&0\\ 0&\sigma_i\sigma_j \end{pmatrix}$$ Now each pauli spin matrix satisfies $\det(\sigma_i)=-1$, so each block matrix lies in $SL_2(\mathbb{C})$. To me this seems to suggest that $\kappa$ is the direct sum of the standard representation on $\mathbb{C}^2$, however I know that this shouldn't be true. Some argument from physics, which I don't follow, seem to suggest that that the representation should be the direct sum of the standard representation and the induced conjugate representation, but I can't see why.
One thing I thought was that it could be that the pauli matrices are a little to nice to figure what the actual representation is. But, I don't really know what else to do.
Any help would be greatly appreciated as it seems that I can't find a succinct answer to this question.