spin homomorphism $SL_2(\mathbb C)\to O(1,3)_e$

Question

I've shown that we have the following action of $SL_2(\mathbb C)$ on $\mathbb R^4$: $$ \phi\colon SL_2(\mathbb C)\to O(1,3) $$ given by $$ \phi_X\colon W\mapsto XWX^\dagger. $$ Since $SL_2(\mathbb C)$ is connected, and our homormophism is continuous, it follows that $\phi(SL_2(\mathbb C))$ is contained in the identity component $O(1,3)_e$. However, I would like to argue that it is equal to $O(1,3)_e$. How can I do that?