Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 2 / 5 / v / Solution

{c}
{title2=$\mathcal B(H_\alpha)=\sum_{\beta\in\Phi}\beta(H_\alpha)\beta$}

Let $\mathcal B(H)=B(H,\cdot)$ be the <Killing form> identification of a <Cartan subalgebra> with its dual. The <root-space decomposition> gives $B(H,K)=\sum_{\beta\in\Phi}\beta(H)\beta(K)$, and therefore the displayed root expansion. On the real <coroot> span its coefficients are real, proving $\mathcal B(\mathfrak h_{\mathbb R})\subseteq\mathbb R P$. For $\mathfrak{sl}_3$ and $H_{12}=E_{11}-E_{22}$, the formula gives $\mathcal B(H_{12})=6(L_1-L_2)$. The normalization is $B(X,Y)=\operatorname{tr}(\operatorname{ad}X\operatorname{ad}Y)$, not a rescaled trace form.

// Target: semisimple-lie-algebra.bigb

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 Incoming links: Killing dual of a coroot