An explicit description of $SL(2,\mathbb{C})$ in terms of $S{O}^{+}(3,1)$ and vice versa

In this note we present explicit and elementary formulas for the correspondence between the group of special Lorentz transformation $S{O}^{+}(3,1)$, on the one hand, and its spin group $SL(2,\mathbb{C})$, on the other hand. Although we will not mention Clifford algebra terminology explicitly, it is hidden in our calculations by using complex $2\times 2$-matrices. Nevertheless, our calculations are strongly motivated by the Clifford algebra $\mathfrak{g}\mathfrak{l}(4,\mathbb{C})$ of four-dimensional space-time.