So I understand that the Lie algebra of $SO(1,3)$ is isomorphic to the Lie algebra of $SU(2)\oplus SU(2)$, and the Lie algebra of $SO(3)$ is isomorphic to one copy of $SU(2)$ (at the group level we have $SO(3)\cong SU(2)/Z_2$).
I guess my question is in two parts:
In ”Physics from Symmetry” by Schwichtenberg, he says that the covering group is more fundamental - i.e. by considering $su(2)$ instead of $SO(3)$ we get spin.
As far as I am aware, if we take the exponential map of a Lie algebra homomorphism/representation then we get a representation of the group. This would imply $SO(3)$ is a representation of $SU(2)$, but how can there exist a homomorphism from $SU(2)$ to $SO(3)$, if each element in $SU(2)$ gets identified with two elements of $SO(3)$?