= Solution
The PDF's specified five-by-five diagonal matrix is $H=H_1$, so $\varepsilon_1(H)=1$ and $\varepsilon_2(H)=0$. In the root sum for the <Killing form>, the two short roots $\pm\varepsilon_1$ contribute $1+1$, the two short roots $\pm\varepsilon_2$ contribute zero, and the four long roots $\pm\varepsilon_1\pm\varepsilon_2$ each contribute one. Therefore
$$
\boxed{B(H,H)=2+0+4=6.}
$$
This computation uses the original PDF matrix; the converted TeX incorrectly gives its dimensions.
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