 A2 fundamental weights and weight lattice

For the A2 root system, ${\omega }_1=(2{\alpha }_1+{\alpha }_2)/3$ and ${\omega }_2=({\alpha }_1+2{\alpha }_2)/3$. The weight lattice $P=\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ contains the root lattice $Q$ with index three. In fundamental coordinates $(a,b)$, root-lattice membership is $a-b\equiv 0(\mathrm{mod}3)$.