I have a set of 45 roots and I want to know which group is generated by the corresponding generators.
In the set are 5 diagonal (=Cartan) generators
$$ (0, 0, 0, 0, 0, 0)_1,(0, 0, 0, 0, 0, 0)_2,(0, 0, 0, 0, 0, 0)_3,(0, 0, 0, 0, 0, 0)_4,(0, 0, 0, 0, 0, 0)_5$$
Therefore the generated group has rank $5$.
16 of the weights are of the form
$$ (1/2,1/2,-1/2,-1/2,-1/2,\sqrt{3}/2) \quad, \quad (-1/2,-1/2,1/2,1/2,1/2,\sqrt{3}/2) \quad , \quad \ldots $$
and 24 are of the form
$$ (0,1,0,1,0,0) \quad, \quad (1,0,0,1,0,0) \quad, \quad (0,0,0,2,0,0) \quad, \quad (1,0,-1,0,0,0) \quad, \quad \ldots $$ How can I determine which group is generated by these roots?
Rank 5 with 45 generators would fit perfectly with $SO(10)$ and indeed some of the roots do coincide with $SO(10)$ roots. Unfortunately, the 16 weights that do look so different are not part of the $SO(10)$ set of roots. Is it $SO(10)$ nevertheless, because of some change of basis?