In the case of SO(3; 1), the first case is excluded since SO(3; 1) is semi-simple. The second case (and the first case) is excluded because SO(3; 1) is simple. For the third case, SO(3; 1) is isomorphic to the quotient SL ( 2 , C ) / { ± ± I } . But { ± ± I } is the center of SL ( 2 , C ) . It follows that the center of SO(3; 1) is trivial, and this excludes the third case. The conclusion is that every representation Π : SO(3; 1) → GL(V) and every projective representation Π : SO(3; 1) → PGL(W) for V, W finite-dimensional vector spaces are faithful.