= Solution
The <weight lattice> is
$$
\boxed{\Lambda_W=\{\lambda\in\mathfrak h^*:\lambda(H_\alpha)\in\mathbb Z\text{ for every root }\alpha\}
=\bigoplus_{i=1}^r\mathbb Z\omega_i,}
$$
where the <fundamental weights> satisfy $\omega_i(H_{\alpha_j})=\delta_{ij}$. The <coroots> use the normalization fixed above.
The <Killing form> defines the complex-linear map
$$
\mathcal B:\mathfrak h\longrightarrow\mathfrak h^*,\qquad
\mathcal B(H)(K)=B(H,K).
$$
By <nondegeneracy of the Killing form on a Cartan subalgebra>, this is an isomorphism. For $H\in\mathfrak h_{\mathbb R}$, expand its image in the dual fundamental-weight <basis>:
$$
\mathcal B(H)=\sum_{i=1}^rB(H,H_{\alpha_i})\omega_i.
$$
The coefficients are real by the preceding positivity calculation, and the $\omega_i$ are in $\Lambda_W$. Consequently
$$
\boxed{\mathcal B(\mathfrak h_{\mathbb R})\subseteq\mathbb R\Lambda_W.}
$$
In fact both spaces have real <dimension> $r$, so this inclusion is equality.
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