  Else from the quotient algebra partition considered in the preceding
episodes, two kinds of partitions on unitary Lie algebras are created by
nonabelian bi-subalgebras. It is of interest that there exists a partition
duality between the two kinds of partitions. With an application of an
appropriate coset rule, the two partitions return to a quotient algebra
partition when the generating bi-subalgebra is abelian. Procedures are proposed
to merge or detach a co-quotient algebra, which help deliver type-AIII Cartan
decompositions of more varieties. In addition, every Cartan decomposition is
obtainable from the quotient algebra partition of the highest rank. Of
significance is the universality of the quotient algebra partition to classical
and exceptional Lie algebras.
