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Identify the Cartan subalgebra with the coordinate space $(t_1,\dots ,t_n)$ used above. The root reflection in any root of a root system fixes the perpendicular hyperplane. For every root of the Cn root system, its complete coordinate action isEvery unlisted coordinate is fixed, and changing the sign of the root leaves its reflection unchanged. These formulas follow directly from $s_{\alpha }(t)=t-\alpha (t){\alpha }^{\vee }$, with the coroot regarded as a vector of the Cartan subalgebra.

The Cn Weyl group is the signed symmetric group: its elements permute the $n$ coordinates and independently change their signs. The Weyl reflections associated with the simple roots interchange consecutive coordinates for $i<n$ and negate the last coordinate for $i=n$. Thus

Identify the diagonal Cartan subalgebra with ${\mathbb{C}}^n$ through its coordinates $(h_1,\dots ,h_n)$, and identify real roots with coordinate vectors using the invariant Euclidean inner product. A Weyl reflection is $s_{\alpha }(h)=h-2(h,\alpha )\alpha /(\alpha ,\alpha )$. It does not change when $\alpha$ is replaced by $-\alpha$. For the Cn root system, its explicit actions areEvery coordinate not displayed stays fixed. Thus each generating Weyl reflection is a signed permutation, so the Weyl group is contained in the signed symmetric group. Conversely, the first row supplies each independent sign change and the second supplies every coordinate transposition; these generate the entire signed symmetric group. This proves the Cn Weyl group identification, including its size: