Question: From Problem 7.11 , suitably normalized simple roots for the algebra (mathrm{G}_{2}) are (alpha_{1}=) ((0, sqrt{3})) and (alpha_{2}=left(frac{1}{2},-frac{sqrt{3}}{2} ight)). What is the corresponding Cartan matrix?

From Problem 7.11 , suitably normalized simple roots for the algebra $\mathrm{G}_{2}$ are $\alpha_{1}=$ $(0, \sqrt{3})$ and $\alpha_{2}=\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$. What is the corresponding Cartan matrix?

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