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In the 3rd episode of the serial exposition, quotient algebra partitions ofrank zero earlier introduced undergo further partitions generated bybi-subalgebras of higher ranks. The refined versions of quotient algebrapartitions admit not only Cartan decompositions of type AI but alsodecompositions of types AII and AIII, resorting to systematic applications ofthe operation tri-addition. Details of quotient algebra partitions of higherranks are extensively examined in this longest episode of the serial.Furthermore, the computational universality is attained taking advantage of aspecial form of transformation called the s-rotation. The structure of quotientalgebra partition is preserved under mappings composed of spinor-to-spinors-rotations.

Quotient Algebra Partition and Cartan Decomposition for su(N) III