I thought that maybe it would be possible to answer this question using the concept of a group presentation.
Let $x_1,x_2,\ldots,x_k$ be $k$ different elements of a group $G$ and $k\geq4$.
If we know that $x_i$ commutes with $x_{i+1}$ and $x_k$ commutes with $x_1$, can we say that all $x_i$ commute with each other ?
Thinking in terms of group presentations, we can ask: does the group
$$\langle x_1, \ldots, x_k \mid (\forall i<k)\ x_ix_{i+1}=x_{i+1}x_i,\ x_kx_1=x_1x_k \rangle$$
satisfy $x_ix_j$ for all $i,j$? In other words, does the normal subgroup generated by $[x_i, x_{i+1}]$ for each $i<k$ and $[x_k, x_1]$ contain $[x_i, x_j]$ for each $i, j$?
But how could we prove or disprove that? Are there any methods that can be used to work out if a certain element is in a certain generated normal subgroup?