Strong associativity of a group algebra

The concept of strongly associative algebra was introduced in Dokuchaev and Exel [Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans Amer Math Soc, 2005] roughly 2 years ago. In Section 3 of Dokuchaev and Exel (to appear), the authors gave an example of a non-strongly associative algebra. Following that example, the strong associativity of the group algebra of the cyclic group of order four was claimed. In this paper, we will prove that this group algebra is not strongly associative.