I am looking for an explicit isomorphism between these two groups:
$$ \mathbb{R}^+ \times {\rm Spin}^c(3,1) \cong {\rm GL}(2,\mathbb{C}) $$
If it makes a difference, I am using this representation for $\mathbb{R}^+ \times {\rm Spin}^c(3,1)$
$$ \mathbf{U} = \exp (A+ F_{01}\gamma_0\gamma_1 + F_{02}\gamma_0\gamma_2+F_{03}\gamma_0\gamma_3 + F_{12}\gamma_1\gamma_2+F_{13}\gamma_1\gamma_3 +F_{23}\gamma_2\gamma_3 + B\gamma_0\gamma_1\gamma_2\gamma_3) $$
how can I link this to ${\rm GL}(2,\mathbb{C})$?
Once the isomorphism is established, my final goal is to eliminate or constrain terms of ${\rm GL}(2,\mathbb{C})$ such that it is reduced to ${\rm GL}^+(2,\mathbb{R})$.
my attempt so far.
Step 1: understand that GL(2,R) can be represent by $\exp(A+X\sigma_x+Y\sigma_y+B\sigma_x\sigma_y)$.
Step 2: To get the representation in step 1, I start with: $$ \mathbf{U} = \exp (A+ F_{01}\gamma_0\gamma_1 + F_{02}\gamma_0\gamma_2+F_{03}\gamma_0\gamma_3 + F_{12}\gamma_1\gamma_2+F_{13}\gamma_1\gamma_3 +F_{23}\gamma_2\gamma_3 + B\gamma_0\gamma_1\gamma_2\gamma_3) $$
I eliminate $F_{03}\to 0,F_{12}\to 0,F_{13}\to 0,F_{23}\to 0$, and pose $\gamma_0\gamma_1\sigma_x, \gamma_0\gamma_2=\sigma_y$. I get:
$$ \mathbf{U} = \exp (A+ F_{01}\sigma_x + F_{02}\sigma_y+ B\gamma_0\gamma_1\gamma_2\gamma_3) $$
At this point, I do not know what to do with $B\gamma_0\gamma_1\gamma_2\gamma_3$. Since this is a pseudo-scalar in 4D, is it sufficient to leave it as is for the 2D case? If not, then I'm afraid I do not know how to convert $B\gamma_0\gamma_1\gamma_2\gamma_3$ to a 2D version $B\sigma_x\sigma_y$.