The spin homomorphism $h:SL(2,\mathbb{C})\to SO(3,1)$, which induces an isomorphism $q:PSL(2,\mathbb{C})\to SO(3,1)^+$, can be described conveniently using the Pauli matrices (see Wikipedia, for example). In particular, it is easy to express the map $h$ explicitly, i.e., to write down the $4\times 4$-matrix in $SO(3,1)^+$ that $h$ assigns to a matrix $SL(2,\mathbb{C})$ with entries $a,b,c,d$. This is done, for example, in Eq. (1) on page 2 of these notes (that reference was simply the first google hit).
However, I could not find a single reference in which such an explicit formula is given for the inverse map $q^{-1}:SO(3,1)^+ \to PSL(2,\mathbb{C})$. By this I mean a formula that says which complex $2\times 2$-matrix (well-defined up to an overall sign) in $PSL(2,\mathbb{C})$ is being assigned by $q^{-1}$ to a given $4\times 4$-matrix in $SO(3,1)^+$. Surely this formula must be found in some textbooks or lecture notes. Who knows a reference?
Of course I know in principle how to deduce the formula but I think it would be a bit tedious and it would also be nice to have a reference to refer to.