Not all nonnegative integer combinations of simple roots are dominant weights

Question

Let $\frak{g}$ be a complex semi-simple Lie algebra with a choice of Cartan, and hence with an associated root system $\Delta$. As is well known, not all nonnegative integer combinations of simple roots $\alpha_i$ are dominant, which is to say, a nonnegative integer combination of fundamental weights $\pi_i$. What are some good/simple examples to illustrate this point?

Answer 1

Consider the irreducible root system of type $G_{2}$ and fix a choice of its simple roots by $\alpha_{1},\alpha_{2}$ such that the Cartan matrix is $$\left(\begin{matrix} 2&-3\\ -1&2 \end{matrix}\right).$$ This matrix tells us neither $\alpha_{1}$ nor $\alpha_{2}$ is a dominant weight, while of course they are nonnegative integer combinations of simple roots!

In general, each off-diagonal entry of the Cartan matrix is non-positive. So all you need to do is to find an irreducible root system having two simple roots that are not orthogonal.