Weyl Chambers of $ B_2$

Question

How many Weyl Chambers/bases does $ B_2$ have?

I thought it was 8, but if instead of for bases using obtuse root pairs you use orthogonal pairs, you get 8 different chambers intersect partially with the chambers associated with obtuse pairs.

This also happens with $ G_2$.

Thanks for the help

Answer 1

If I am not mistaken, the type $B_{2}$ corresponds to Lie algebra $\mathcal{so}(3)$. Its Weyl group should have order $$2\times 6=12$$ as it should be isomorphic to $\mathbb{Z}_{2}\rtimes S_{3}$. So the Weyl chamber should have equally many chambers.