Solution
= Solution
The <fundamental weights> satisfy $2(\omega_i,\alpha_j)/(\alpha_j,\alpha_j)=\delta_{ij}$. Since $|\alpha_1|^2=1$ and $|\alpha_2|^2=2$, solving gives
$$
\omega_1=\left(\frac12,\frac12\right),
\qquad \omega_2=(0,1).
$$
The full <B2 root system> is
$$
\{(\pm1,0),(0,\pm1),(\pm1,\pm1)\}.
$$
The <weight lattice> is generated by $\omega_1,omega_2$; geometrically it consists of the integer lattice together with the translate in which both coordinates are half-integers.