$SL(2,\mathbb{C})\cong \operatorname{Spin}(1,3,\mathbb{R})?$ or $SL(2,\mathbb{C})\cong \operatorname{Spin}(3,1,\mathbb{R})?$

Question

Which one of the isomorphism is correct? $$SL(2,\mathbb{C})\cong \operatorname{Spin}(1,3,\mathbb{R})?$$ $$SL(2,\mathbb{C})\cong \operatorname{Spin}(3,1,\mathbb{R})?$$

If $SL(2,\mathbb{C})\cong \operatorname{Spin}(1,3;\mathbb{R})$, then what would another interpretations of the isomorphism $\operatorname{Spin}(3,1,\mathbb{R}) \cong ?$

Vice versa.

See also https://math.stackexchange.com/a/3027018/141334

Answer 1

Indeed, we always have $$ {\rm Spin}(p,q,\Bbb R)\cong {\rm Spin}(q,p,\Bbb R), $$ see for example wikipedia, or nLab.

For the orthogonal groups we have the isomorphism $$ f\colon O(p,q)\rightarrow O(q,p),\; A \mapsto \phi\circ A\circ \phi^{-1}, $$ where $\phi\colon \Bbb R^{p+q}\rightarrow \Bbb R^{q+p}$ is the so-called anti-orthogonal map.