Expansion of Lie Groups and Representations of SL(3, C)

We show that in any unitary representation of SL(3, C) belonging to the principal series, the generators can be expressed as functions of the generators of a suitably chosen unitary representation of the non-semisimple group SU(3) × T8. Both the nondegenerate and the degenerate series are considered. Since the representations of SU(3) × T8 in an SU(3) basis are known, this provides a solution to the multiplicity problem in setting up the SL(3, C) representations in an SU(3) basis.