The generator set of a Lie algebra is, so to say, a rank 3 tensor with 1 upper and 1 lower index running in the defining irrep and one lower running in the adjoint irrep. The structure constants C are a rank 3 tensor with indices only running in the adjoint irrep. (This is spoken very graphic-oriented, replace with clean vector-space notation if it makes you feel better :-) What happens if one replaces the two instances of "defining" in the above sentence with two instances of "any irrep" R? The dimensions change, but the structure constants stay the same (or at least, you can choose a gauge where they do)? So still [Ri,Rj]=sumCijk*Rk?
At least the standard SU2 example affirms my hunch - the rules are always [Jx,Jy]=Jz regardless of the spin (= the irrep), and graphically, this means that the adjoint J=1 is in any Clebsch j(x)j (except J=0 of course), which is obviously true. (Taking R to be the adjoint even forces more cool stuff, IHX rule and so, but enough for now.)
Is my intution right? -- Hauke Reddmann <:-EX8 firstname.lastname@example.org Hund frißt Hund jeden Tag - Pal jetzt NEU mit Menschgeschmack Hund frißt Hund heißt der Sport - hoff', du stehst auf Völkermord (Der Nachwuchs)