= Solution
Let $E$ be the real span of the roots in $\mathfrak h^*$. The <weight lattice> is
$$
\boxed{\Lambda_W=P=\{\mu\in E:\mu(h_\alpha)\in\mathbb Z\text{ for every root }\alpha\},}
$$
where $h_\alpha$ is the <coroot> in the root's <sl2 subalgebra associated with a root>, normalized by $\alpha(h_\alpha)=2$. It suffices to test the simple coroots. Equivalently, if $\omega_i(h_{\alpha_j})=\delta_{ij}$ defines the <fundamental weights>, then $P=\bigoplus_i\mathbb Z\omega_i$.
To prove <integrality of representation weights on coroots>, restrict a finite-dimensional module $V$ to the root's <sl2 Lie algebra>. By the <classification of finite-dimensional sl2 representations>, the eigenvalues of its Cartan element $h_\alpha$ are integers. Thus every $\mu\in\Lambda_V$ has $\mu(h_\alpha)\in\mathbb Z$. The simple coroots form a basis of the real Cartan space, so these integral values also put $\mu$ in $E$. Therefore $\boxed{\Lambda_V\subseteq\Lambda_W}$ for every finite-dimensional representation in this setting. The weight lattice need not coincide with the smaller <root lattice>.
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