Isomorphism of the special orthogonal group

I want to determine whether $SO(4,\mathbb{C})=SL_2(\mathbb{C})\times SL_2(\mathbb{C})$ as Lie groups.

I have tried proving that they are not isomorphic by looking at simply connectedness. However, I am stuck.

Any help would be appreciated.

Thank you!

Solution 1:

Hint: We have $$ SO(4, \Bbb C)=(SL(2, \Bbb C)×SL(2,\Bbb C))/\Bbb Z_2 $$ which follows the same way as the proofs in these posts

How to visualize $SO(4) \simeq SO(3)\bigotimes SO(3) / \mathbb{Z}_2 $

Why isn't $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

The Lie algebras however are isomorphic, and also $$ \mathfrak{so}_4(\Bbb C)\cong \mathfrak{so}_3(\Bbb C)\oplus \mathfrak{so}_3(\Bbb C)\cong \mathfrak{sl}_2(\Bbb C)\oplus \mathfrak{sl}_2(\Bbb C). $$