It is well known that if $\alpha$ and $\beta$ are a set of simple roots for the root system of type $A_2$ and if the angle between them is $\frac{\pi}{3}$, then the Cartan integers $\langle \alpha, \beta \rangle = \langle \beta, \alpha \rangle = 1$. How does this agree with the fact that every couple of elements $\alpha_i, \alpha_j$ belonging to a base of a root system must be such that $(\alpha_i, \alpha_j) \leq 0$?
I came up with this question by looking at the Cartan matrix and the associated 2-dimensional representation of $A_2$, where of course the angle between $\alpha$ and $\beta$ is obtuse. Why do the non-diagonal entries of the matrix always have to be negative? In other words, is it because the scalar product between two basis elements must be negative or because it is just a standard form?