There must something wrong with my understanding of this relations because I think the usual way they are derived in many textbooks makes no sense. It goes like this, first assume that to every rotation O(a) in euclidean space there exists a rotation operator R(a) in Hilbert space,second: the relation stated first is an homomorphism T,that is T(O)=R So far, so good the problem is that after verifying the relation O_x(da)O_y(db)-O_y(db)Ox(da)=O_z(dadb)-I, for infinitesimal rotations da and db in euclidean space the authors conclude that a similar relation holds for infinitesimal rotations in Hilbert space. This last step requieres that besides being T(O_1O_2)=T(O_1)T(O_2) which is ok, the relation T(O_1+O_2)=T(O_1)+T(O_2) must also hold. is it correct what I'm saying? in which case, why should T(O_1+O_2)=T(O_1)+T(O_2) hold?