I have seen sources claim that $SO^+(1,3) \cong SU(2) \times SU(2)$, but have seen others claim that only their Lie algebras are isomorphic.
- Is it true that $SO^+(1,3) \cong SU(2) \times SU(2)$?
- If not, is $SO^+(1,3)$ isomorphic to some quotient of $SU(2) \times SU(2)$?
- Is the analogous result true for their Lie alebras, i.e. $\mathfrak{so}^+(1,3) \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$, or something similar?
- Generally, when can you go from a product group isomorphism to a corresponding result for Lie algebras, or vice versa?
I'm led to believe there is some isomorphism of this form since (I think) $$SO^+(1,3) \cong PSL(2,\mathbb{C}), \quad SL(2,\mathbb{C}) \cong SO(4), \quad SO(4) \cong SU(2) \times SU(2)/\{\pm I\}.$$