I learnt that the Coxeter groups have a few members more than the classic simple Lie groups: $H_3, H_4$ and $I_2(p)$. Is there a Reshetikhin-Turaev invariant for those, too? If not, where does the construction fail (maybe there is not even an associated quantum group)?
BTW, you would already help me by filling out the gap:
$I_2(5): R\bigotimes R=...\bigoplus ...$ (also for the defining irrep R)