Let $x,y,z$ be dimensions that appear in the Clebsch-Gordan series
$x\*x=1+t+u+y+z$. 
(E8 family if $t=x$ (say), but there is at least another family. E.g. B4(R4) belongs to the latter.) 


With the right pick of dimension ($t,u,y,z$ are not equivalent!) I got the following diophantine equation: 


$$\eqalign{-27(-1+x)x^2(2+x)^2y+54x(2+x)^2y^2+9(-1+x)x^2(2+x)^2z\cr-24(-4+x)(-1+x)x(2+x)yz
+16(-1+x)(2+x)(8+x)y^2z\cr-18x^2(2+x)z^2+48(-4+x)xyz^2-32(8+x)y^2z^2=0\cr}$$ 


(No rational solutions please, as these are dimensions. I don't exclude negative $x,y,z$ for now, though.)  

Solving after $y$, $$x(36x+36x^2+9x^3-32z+16xz+16x^2z-32z^2)=a^2$$ ($a$ is still integer) and solving that after $z$, $$2x^2(2+x)^3(1+2x)=2a^2x+b^2$$ ($b$ is also integer.) 


I'm stuck here. Were this a hunt for rational solutions, I'd set $a=x(x+2)c$ and $b=x(x+2)d$ and use the standard method for Pythagorean triangles afterwards. But I'm not sure $c$ and $d$ are integers (even constraining to the actual solutions like E8(R1),...), and the Pythagorean parametrization usually also is done with rationals. 


Can you still give a parametrization $x=f(p,q)$, $y=g(p,q)$, $z=h(p,q)$ with integer $p,q$? (As I said, with rational $p,q$ this is trivial, in fact I started from that!) This should be elementary number theory...but it's already too high for me. 



A diophantine equation for the E8 knot polynomial family
You will find some information on this in my papers:


MR1960703 (2004g:17003a) Westbury, Bruce W.
$R$-matrices and the magic square.
 J. Phys. A 36 (2003), no. 7, 1947--1959.


MR2029689 (2005g:18014) Westbury, Bruce .
Invariant tensors and diagrams.
Proceedings of the Tenth Oporto Meeting on Geometry, Topology and
 Physics (2001).
 Internat. J. Modern Phys. A 18 (2003), October, suppl., 49--82.


