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:
(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.