 Root hyperplane

The hyperplane perpendicular to a root of a root system is its root hyperplane, interpreted as the kernel of the associated linear functional on the Cartan space. The corresponding reflection is $s_{\alpha }(v)=v-2⟨v,\alpha ⟩\alpha /⟨\alpha ,\alpha ⟩$. For the B2 root system, the four distinct hyperplanes are the two coordinate axes and the two diagonals; their reflections generate the signed-permutation Weyl group of order eight.