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How do you know that this is possible to do?

Some 4x4 matrices are not block diagonalizable into 2x2 blocks. For example a nilpotent matrix with a singe Jordan block. If you know for some reason that your symbolic matrix is diagonalizable into 2x2 blocks then probably there is a way to do this, but I don't think possible to write an algorithm that can decide if a symbolic matrix is block diagonalizable.