The map P is a group homomorphism into GL ( h ) ⊂ ⊂ End ( h ) . Thus P : SL ( 2 , C ) → → GL ( h ) is a 4-dimensional representation of SL ( 2 , C ) . Its kernel must in particular take the identity matrix to itself, A IA = A A = I and therefore A = A. Thus AX = XA for A in the kernel so, by Schur's lemma, A is a multiple of the identity, which must be ± I since det A = 1. The space h is mapped to Minkowski space M, via