In [the reviewer et al., Commun. Algebra 45, No. 12, 5212–5226 (2017; Zbl 1425.11067)] it was proven that if \(D_1=(\alpha,\beta)_{2,F}\) and \(D_2=(\alpha,\gamma)_{2,F}\) are quaternion algebras over a field \(F\) of characteristic not 2 sharing a common slot, then \((\alpha) \cup (\beta) \cup (\gamma)\) is a well-defined invariant of the pair in \(H^3(F,\mu_2)\), i.e., for any \(a,b,c \in F^\times\) for which \(D_1=(a,b)_{2,F}\) and \(D_2=(a,c)_{2,F}\), \((a) \cup (b) \cup (c)=(\alpha) \cup (\beta) \cup (\gamma)\). It is natural to ask whether this extends to cyclic algebras of degrees greater than 2, and the answer is evidently no, as the author of this current paper demonstrates: if \(D_1=(\alpha,\beta)_{n,F}\) and \(D_2=(\alpha,\gamma)_{n,F}\), then also if \(D_1=(\alpha^{-1},\beta^{-1})_{n,F}\) and \(D_2=(\alpha^{-1},\gamma^{-1})_{n,F}\). If the invariant were well-defined, we would get \((\alpha) \cup (\beta) \cup (\gamma)=(\alpha^{-1}) \cup (\beta^{-1}) \cup (\gamma^{-1})\). But since \(-(\alpha) \cup (\beta) \cup (\gamma)=(\alpha^{-1})\), when \(n\) is odd, this would mean that \((\alpha) \cup (\beta) \cup (\gamma)\) must be trivial, and it does not have to be (e.g., the generic case). The author uses the same example to show that the graph of the different pairs of symbol presentations \((\alpha,\beta)_{n,F},(\alpha,\gamma)_{n,F}\) of \(D_1,D_2\), with edges between two pairs when the symbol modification is as small as possible, is not connected, which disproves the “chain lemma” in this case.