Suppose now that the Lie algebra formula_3 is the Lie algebra of a Lie group "G". Then we say that formula_46 is analytically integral ("G-integral") if for each "t" in formula_4 such that formula_72 we have formula_73. The reason for making this definition is that if a representation of formula_3 arises from a representation of "G", then the weights of the representation will be "G"-integral. For "G" semisimple, the set of all "G"-integral weights is a sublattice "P"("G") ⊂ "P"(formula_3). If "G" is simply connected, then "P"("G") = "P"(formula_3). If "G" is not simply connected, then the lattice "P"("G") is smaller than "P"(formula_3) and their quotient is isomorphic to the fundamental group of "G".