= Solution
Work with a complex <semisimple Lie algebra> and its <root system> $R$ in the real span $E$ of the <roots of a root system>. The <Killing form> induces a positive definite <inner product> on this real root space, after the usual identification with the dual of the real span of the <coroots>. For each root $\alpha$, its <Weyl reflection> is
$$
s_\alpha(\lambda)=\lambda-\langle\lambda,\alpha^\vee\rangle\alpha,\qquad \alpha^\vee=\frac{2\alpha}{(\alpha,\alpha)}.
$$
The <Weyl group> is $\boxed{W=\langle s_\alpha:\alpha\in R\rangle}$. The structural facts being used are that these <reflections> preserve the finite <root system>, that the roots span $E$, and that the reflections in the <simple roots> already generate $W$. Thus $W$ is finite: it acts faithfully as permutations of $R$. Equivalently, it is the normalizer of the <Cartan subalgebra> modulo its centralizer in the associated adjoint group.
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