We prove that the equational theory of Kleene algebra with commutativity conditions on atomic terms is undecidable, thereby settling a longstanding open question in the theory of Kleene algebra. In fact, we show that this undecidability result holds even if we drop the induction axioms of Kleene algebra, which leads to a simpler equational theory. This complements a recent result of Kuznetsov, who independently established a similar undecidability result, but relying on induction.