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In this note we present explicit and elementary formulas for thecorrespondence between the group of special Lorentz transformation $SO^+(3,1)$,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 ishidden in our calculations by using complex $2\times2$-matrices. Nevertheless,our calculations are strongly motivated by the Clifford algebra$\mathfrak{gl}(4,\mathbb{C})$ of four-dimensional space-time.

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