Root Reflections Lie in the Weyl Group identifies the role of the Weyl group in integration, chambers, and conjugacy. In concrete terms, it formalizes the statement that let be a compact connected Lie group with gT G t X^(T) g CT X^(T)de: t i R t C C t C(H)=2s: t C^ t C^s()=-(H) NG(T)(n)(t)=(n^-1tn)X^(T)s W(G,T):=NG(T)/Ts()=$..