Long-root A2 subsystem of G2 ($A_2\subset G_2$)

For short and long simple roots $\alpha ,\beta$ of $G_2$, the long roots form an $A_2$ root system with simple roots ${\delta }_1=\beta$ and ${\delta }_2=3\alpha +\beta$. If ${\omega }_1,{\omega }_2$ are the corresponding $A_2$ fundamental weights, then $\alpha ={\omega }_2-{\omega }_1$ and ${\alpha }^{\vee }={\delta }_2^{\vee }-{\delta }_1^{\vee }$.

The seven-dimensional fundamental $G_2$ representation has weights zero and the six short roots. Under its long-root $A_2$ subalgebra, these are the weights of the defining three-dimensional representation, its dual, and the trivial representation: