In my hunt for spurious "alternatives" to the $E_7$ family I always encounter
"fake" solutions. They turn out to be mostly $E_7$ family solutions disguised
by $q\rightarrow{i*q}$. The effect is that the dimension (at $q=1$) comes out totally
wrong. Generalizing a bit:
Take a random quantum dimension, say $d(G_2)=1+q^2+1/q^2+q^8+1/q^8+q^{10}+1/q^{10}$
and plug in a random root-of-unity $q=(-1)^{m/n}$. (Since even I know that the
"interesting" q are these.) For small $n$ almost all values of $d$ will be integer.
1. This surely has to do with the fact that $d(q)$ is a ratio of quantum
integers, i.e. cyclotomic?
2. Do these pseudo-dimensions have some "intuitive" meaning?