I am trying to show that for translations in 3D, the commutation relation between a component of position $\mathbf{X}$ and a component of the translation generator $\mathbf{P}$ is given by
$$[X_i,P_j]=i \hbar \delta_{ij}\mathbf{I}$$
where $\mathbf I$ is the identity operator.
According to my lecturer, this can be derived by expanding the translation operator for an infinitesimal translation $\delta \mathbf{a}$ as
$$U(\delta \mathbf{a})=\mathbf{I}-i \delta \mathbf{a} \cdot \mathbf{P}/\hbar$$ to first order, then writing $$U^\dagger \mathbf{X} U=\mathbf{X}+\delta \mathbf{a}$$ and equating coefficients of $\delta \mathbf{a}$ to obtain $$\frac{i}{\hbar}[\delta \mathbf{a}\cdot \mathbf{P},\mathbf{X}]=\delta \mathbf{a}. \hspace{1cm}(*)$$ My lecturer then claims that since this result above is true for arbitrary 3D infinitesimal translations $\delta \mathbf a$, it must also be true for the Cartesian components of the translation generator and hence it follows that
$$[X_i,P_j]=i \hbar \delta_{ij}\mathbf{I}.$$
I’m not sure how this follows from $(*)$, however. How are you meant to go from the dot product of operators to considering the Cartesian components themselves? It would be helpful if someone could spell this out more directly.
