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Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...

There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...