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Relation between the root lattice of $\mathrm{SO}(7)$ and the root lattice of $G_2$

The root lattice of $\mathfrak{so}(7)$ is given by the following 18 roots: $$ \left(\begin{array}{c}0\\0\\1\end{array}\right) , \left(\begin{array}{c}0\\0\\-1\end{array}\right) , \left …
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