Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 3 / v / Solution

The PDF's specified five-by-five diagonal matrix is $H=H_1$, so ${\epsilon }_1(H)=1$ and ${\epsilon }_2(H)=0$. In the root sum for the Killing form, the two short roots $\pm {\epsilon }_1$ contribute $1+1$, the two short roots $\pm {\epsilon }_2$ contribute zero, and the four long roots $\pm {\epsilon }_1\pm {\epsilon }_2$ each contribute one. ThereforeThis computation uses the original PDF matrix; the converted TeX incorrectly gives its dimensions.