In Sakurai Modern Quantum Mechanics I saw the author takes commutation of two infinitesimal 3D rotation matrices. He also claims that the Hilbert space rotation operators should satisfy the same commutation relation.
What does it mean to subtract two elements in the representation of a group, namely, lets say $R(ε)$ to be infinitesimal matrix by infinitesimal angle $ε$ then what does $$ R_x(ε)R_y(ε) - R_y(ε)R_x(ε) = R_z(ε^2)-1 $$ mean? How can we subtract rotations from each other because rotations form a group in which multiplication operation is defined (applying rotations consecutively). But we do not have any addition operation in rotation group, so how do we have such an operation in the representation of the group and what does it mean such things like the infinitesimal rotation commutations are related to second order infinitesimal rotations around some other axis.
He also claims that the Hilbert space rotation operators should satisfy the same commutation relation. How can he be sure about this statement