It is well known that $SL_2(\mathbb{C})$ is the universal cover of $SO^+(1,3)$, see for example the Wikipedia page on the Lorentz group 1. The map goes like this (some of this is not standard notation):
First, establish a bijection between $\mathbb{R}^{1,3}$ and the set $M_{s.a.}(2,\mathbb{C})$ of self-adjoint 2x2 matrices with complex entries via
\begin{align*} \sigma:\mathbb{R}^{1,3}&\to M_{s.a.}(2,\mathbb{C}) \\ (x_0,x_1,x_2,x_3)&\mapsto \begin{pmatrix}x_0+x_3&x_1-ix_2 \\ x_1+ix_2 & x_0-x_3\end{pmatrix} \end{align*}
An important property of $\sigma$ is that it preserves the metric in the sense that \begin{equation} \mathrm{det}(\sigma(x))=x^2:=x_0^2-x_1^2-x_2^2-x_3^2 \end{equation}
Then there is a natural action of $SL_2(\mathbb{C})$ on $M_{s.a.}(2,\mathbb{C})$ by \begin{align*} \lambda:SL_2(\mathbb{C})&\to \big(M_{s.a.}(2,\mathbb{C})\to M_{s.a.}(2,\mathbb{C})\big) \\ A&\mapsto (X\mapsto AXA^*) \end{align*}
The determinant property guarantees that it is well-defined. Then the so-called spinor representation is the induced map
\begin{align*} \Lambda:SL_2(\mathbb{C})&\to \big(\mathbb{R}^{1,3}\to\mathbb{R}^{1,3}\big) \\ A&\mapsto \sigma^{-1}\circ\lambda(A)\circ\sigma \end{align*}
It is then shown that the $\mathrm{Im}(\Lambda)=SO^+(1,3)$ and $\ker(\Lambda)=\{\pm1\}$. The explicit matrix elements of $\Lambda(A)$ can be calculated in terms of taking traces of various combinations of Pauli matrices (see the wiki for details). The 1st isomorphism theorem then implies that the quotient group $PSL_2(\mathbb{C}):=SL_2(\mathbb{C})/ \{\pm 1\}$ is isomorphic to the image of $\Lambda$ \begin{align*} \tilde{\Lambda}:PSL_2(\mathbb{C})&\to SO^+(1,3) \\ [A]&\mapsto \Lambda(A) \end{align*}
My question is: How can we describe the reverse map $\tilde{\Lambda}^{-1}(L)=\{\pm A_L\}$? This is given as an exercise on pg 134 of this book
Bogoljubov, Nikolaj N.; Logunov, Anatolij A.; Todorov, Ivan T., Introduction to axiomatic quantum field theory. Authorized translation from the Russian original by Stephen A. Fulling and Ludmilla G. Popova. Edited by Stephen A. Fulling, Mathematical Physics Monograph Series 18. Reading, MA: Benjamin. xxvi, 707 p. (1975). ZBL1114.81300.
Is there an explicit formula for the 2x2 matrix in the fiber above a Lorentz transform in terms of the its matrix elements? We know that $\tilde{\Lambda}$ is invertible for abstract reasons, but I was hoping for something a bit more concrete. Thanks!
