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In the book by V.Kac "Infinite dimensional Lie algebras" in proposition 4.9 it is argued that Weyl group constructed by Cartan matrix which defines positive-definite metric on Cartan subalgebra is finite. The way argument is presented in in the book: Weyl group is a subgroup of compact orthogonal group because the metric is positive-definite. Also Weyl group preserves the root lattice, hence it is discrete. This implies that the Weyl group is finite.

Why does it imply? Any discrete subgroup of orthogonal group has to be compact? What about subgroup of SO(2) generated by an element: rotation by an angle 2 pi x, where x is a non-zero irrational number? Is not it counterexample?

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That is not a counterexample, because the subgroup is not discrete. – Mariano Suárez-Alvarez Oct 18 '11 at 3:40
Your question is off-topic here, as explained in the FAQ. is a good place, to ask, though. – Mariano Suárez-Alvarez Oct 18 '11 at 3:41
"discrete" here means "is discrete with the induced topology from the ambient Lie group". If you think about the irrational angle, you'll see that what you've done is given a homomorphism from the discrete group $\mathbb Z$ into $\mathrm{SO}(2)$, but the image is not discrete as a topological space. – Theo Johnson-Freyd Oct 18 '11 at 4:23

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