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knowwhatyoudontknow

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- TL;DR Summary
- Trying to understand the process for obtaining the block diagonal form of the rotation matrix for the point group C3V.

I am looking at the point group C<sub>3v</sub> described shown here. I am trying to understand the block diagonalization process. The note says that changing the basis in the following way will result in the block diagonal form.

What is the rationale for choosing the new basis. Is it chosen by trial and error or is there some method behind it? I know how to generate the COB matrix using the Clebsch-Gordan coefficients in the case of angular momentum but just can't make the transition in this case. Any help would be appreciated.

What is the rationale for choosing the new basis. Is it chosen by trial and error or is there some method behind it? I know how to generate the COB matrix using the Clebsch-Gordan coefficients in the case of angular momentum but just can't make the transition in this case. Any help would be appreciated.