= Solution
For an arbitrary finite-dimensional complex <Lie algebra> $L$, a <Cartan subalgebra> is a nilpotent <Lie subalgebra> $H$ equal to its own <normalizer of a Lie subalgebra>[normalizer]. When $L$ is <Semisimple Lie algebra>[semisimple], this is equivalently a maximal abelian subalgebra consisting of elements that act semisimply in the <Adjoint representation>.
Choose such an $H$. Simultaneous diagonalization gives the <root-space decomposition>
$$
L=H\oplus\bigoplus_{\alpha\in\Phi}L_\alpha,
\qquad
L_\alpha=\{x\in L:[h,x]=\alpha(h)x\text{ for every }h\in H\}.
$$
The nonzero <weight of a representation>[weights] $\alpha\in H^*$ are the roots. The restriction of the <Killing form> $B$ to $H$ is a <nondegenerate bilinear form>, so each $\alpha$ corresponds to a unique $t_\alpha\in H$ with $\alpha(h)=B(t_\alpha,h)$. On the real span of these $t_\alpha$, the restriction of $B$ supplies a positive-definite inner product after choosing the standard real form. The <sl2 subalgebra associated with a root> gives
$$
s_\alpha(\beta)=\beta-\langle\beta,\alpha^\vee\rangle\alpha,
\qquad
\langle\beta,\alpha^\vee\rangle\in\mathbb Z,
$$
and shows that these reflections preserve the finite set $\Phi$. Thus the roots form a finite reduced crystallographic <root system>, whose <Weyl group> is generated by these reflections.
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