Free presentations of Weyl groups

Can someone please provide me with a reference for the fact that the Weyl group $$W$$ associated to a root system $$\Phi$$ can be realised as a Coxeter group?

This means that a Weyl group $$W$$ has a presentation of the form $$\langle s_i \mid s_i^2, (s_is_j)^{n_{ij}}\rangle.$$

This is normally done by choosing a set of positive roots $$\Phi^+$$ and a set of simple roots $$\Sigma=\lbrace \alpha_i \rbrace \subset \Phi^+$$ and mapping the generators of the free group $$\langle s_i\rangle$$ to the $$s_{\alpha_i}$$, the reflections across the hyperplane perpendicular to each $$\alpha_i$$. If we choose the $$n_{ij}$$ to be the order of $$s_{\alpha_i}s_{\alpha_j}$$ then clearly this map factors to the quotient.

The difficulty is in showing that the map is injective. If we happen to have $$s_{i_1}\dots s_{i_n}$$ and $$s_{j_i}\dots s_{j_m}$$ mapping to the same element $$w$$, we must show that the former elements are the same in the quotient of the free group. I have a vague understanding that this can be done by looking at the two induced 'galleries' connecting the fundamental domain (Weyl chamber)) $$C$$ with $$wC$$.

I hope someone can provide a reference with all the details of the argument in the preceeding paragraph (preferably not Bump's Lie Groups 2nd Ed.).

A proof can be found in Section 1.9 of James E. Humphreys’ Reflection groups and Coxeter groups. However, this proof does not use Weyl chambers but instead argues that any relation $$s_{\alpha_{i_1}} \dotsm s_{\alpha_{i_n}} = 1$$ in $$W$$ can be deduced from the given relations $$(s_{\alpha_i} s_{\alpha_j})^{n_{ij}} = 1$$.