Coverings of the Lorentz group

Question

I noticed the followings:

1. $SL(2,\mathbb{C})$ is a double cover of $SO(1,3)$.

2. $SL(2,\mathbb{C})$ is a double cover of $PSL(2,\mathbb{C})$

3. $PSL(2,\mathbb{C})$ is isomorphic to $SO^+(1,3)$ which is the restricted Lorentz group.

I would expect that $PSL(2,\mathbb{C}) \cong SO(1,3)$, not $PSL(2,\mathbb{C}) \cong SO^+(1,3)$ since $SO(1,3)$ is not isomorphic to $SO^+(1,3)$.

Actually I think $SO(1,3)$ should be a 4-fold cover of $SO^+(1,3)$.

Can someone shed some light on this?

Answer 1

$SL(2,\mathbb{C})$ is the double cover of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$, not the proper Lorentz group $SO(1,3;\mathbb{R})$. See also e.g. this Phys.SE post. Be aware that notation and terminology may differ for different authors.