Roots of a Compact Connected Lie Group Form a Crystallographic Root System identifies how roots and root data encode compact connected Lie groups. In concrete terms, it formalizes the statement that let be a compact connected Lie group with gT GG tS^1:=\z C:|z|=1\ g C:= g R C t C:= t R C X^(T):=(T,S^1) Re Tde: t i ReE:= R()E1 induced from any -invariant [inner product](/page/Inner%20Product) on restri.