= Solution
Use the standard <coroot> normalization $\alpha(H_\alpha)=2$. The symbols $H_\alpha$ are vectors in the <Cartan subalgebra>, not the root-space vectors $X_\alpha$. The <root-string theorem> gives $\beta(H_\alpha)\in\mathbb Z$, so every <root> takes real values on
$$
\mathfrak h_{\mathbb R}=\operatorname{span}_{\mathbb R}\{H_\beta:\beta\in\Phi\}.
$$
The simple <coroots> form a <basis> of this real space.
In the <root-space decomposition>, $\operatorname{ad}H$ is zero on $\mathfrak h$ and <scalar multiplication> by $\beta(H)$ on the one-dimensional space $\mathfrak g_\beta$. Therefore, for $H,K\in\mathfrak h$,
$$
B(H,K)=\sum_{\beta\in\Phi}\beta(H)\beta(K).
$$
For $H,K\in\mathfrak h_{\mathbb R}$ every term is real, so the restricted form is real. Moreover,
$$
B(H,H)=\sum_{\beta\in\Phi}\beta(H)^2\geq0.
$$
If it is zero, all $\beta(H)$ vanish. Then $H$ commutes with every <root space> and with $\mathfrak h$, hence is central in $\mathfrak g$. The <center of a Lie algebra> vanishes for a <semisimple Lie algebra>, so $H=0$. Thus the <Euclidean subspace of a Cartan subalgebra> has
$$
\boxed{B|_{\mathfrak h_{\mathbb R}}\text{ real and positive definite}.}
$$
Equivalently, positivity follows because the <roots> span $\mathfrak h^*$.
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