Lie group geometry, Clifford algebra, symmetric spaces, Kaluza-Klein, and all that
The kernel of this map from SU(2) to SO(3) is equal to the set of elements of SU(2) that are mapped into the identity element of SO(3). So, yes, these are the elements 1 and -1 of SU(2).
Heh Mehdi, want to take a shot at calculating the Killing vector fields corresponging to the right action of the su(2) Lie generators? Joe almost got them right, but we haven't heard from him in awhile...