Commutation relation of the generators of Lorentz Group

Question

$$\left[J_i,J_j \right]=i\epsilon_{ijk}J_k$$ $$\left[J_i,M_j \right]=i\epsilon_{ijk}M_k$$ $$\left[M_i,M_j \right]=-i\epsilon_{ijk}J_k$$ where $J_i$ is the generator of rotation of Lorentz group, $M_i$ is the generator of boost of Lorentz group.

Can somebody give me a physical reason for why the commutation relation for $\left[J_i,M_j \right]$ is not zero. I think it's not zero because a rotation in one inertial frame is not equivalent to a rotation in another inertial frame. Am I correct?

Answer 1

The physical reason is that the boost $K_j$ behaves as a vector under $SO(3)$ rotations (which in turn are generated by $J_i$), i.e. $$[J_i, K_j]~=~ i\epsilon_{ijk} K_k. $$