Let us consider the Lorentz group $SO(1, 3)$. There are two copies of $SU(2)$ in it. We specify a matrix representation of $SO(1, 3)$ by a doublet $(j, j')$; where $j$ corresponds to the SU(2) generated by $N_i^+$ and $j'$ corresponds to the SU(2) generated by $N_i^-$; $N_i^\pm$ are generators of the $SU(2)$s.

Say, I want to write a matrix element of the $(0, \frac{1}{2})$ representation of $SO(1, 3)$. How can I mathematically construct such an element? A detailed discussion would be highly appreciated.