In my note it says for Poincare transform $\delta x^\mu=a^\mu+\omega^\mu_\nu x^\nu$: to derive the commutator relation of algebra, we consider 2 consecutive transforms $$\delta_2\delta_1 x^\mu=(\omega_1)^\mu_\nu(a_2)^\nu+(\omega_1)^\mu_\lambda(\omega_2)^\lambda_\nu x^\nu$$
but it does not look correct to me.
I have worked out $$\delta_2(a_1^\mu+(\omega_1)^\mu_\nu x^\nu)=a_2^\mu+(\omega_2)^\mu_\lambda(a_1^\lambda+(\omega_1)^\lambda_\nu x^\nu)=a_2^\mu+(\omega_2)^\mu_\lambda a_1^\lambda+(\omega_2)^\mu_\lambda(\omega_1)^\lambda_\nu x^\nu$$
Thus the commutator should be $$(\delta_1\delta_2-\delta_2\delta_1)x^\mu=a_1^\mu-a_2^\mu+(\omega_1)^\mu_\lambda a_2^\lambda-(\omega_2)^\mu_\lambda a_1^\lambda+(\omega_1)^\mu_\lambda(\omega_2)^\lambda_\nu x^\nu-(\omega_2)^\mu_\lambda(\omega_1)^\lambda_\nu x^\nu$$ which is noticeably different from the note. I'm not sure where it went wrong?
Also an additional question: in general (regarding Lorentz/Poincare transformations), how do we derive commutation relations of generators $P_\mu,M_{\mu\nu}$ etc. from the above $(\delta_1\delta_2-\delta_2\delta_1)x^\mu$?
