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