Let FG denote the group algebra of the group G over the field F with char(F)≠2. Given both a homomorphism σ:G→{±1} and a group involution ∗:G→G, an oriented involution of FG is defined by α=Σαgg↦α⊛=Σαgσ(g)g∗. In this article, we determine the conditions under which the group algebra FG is normal, that is, conditions under which FG satisfies the ⊛-identity αα⊛=α⊛α. We prove that FG is normal if and only if the set of symmetric elements under ⊛ is commutative.