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We prove that if a right-angled Artin group $A_\Gamma$ is abstractlycommensurable to a group splitting non-trivially as an amalgam or HNN-extensionover $\mathbb{Z}^n$, then $A_\Gamma$ must itself split non-trivially over$\mathbb{Z}^k$ for some $k\le n$. Consequently, if two right-angled Artingroups $A_\Gamma$ and $A_\Delta$ are commensurable and $\Gamma$ has noseparating $k$-cliques for any $k\le n$ then neither does $\Delta$, so"smallest size of separating clique" is a commensurability invariant. We alsodiscuss some implications for issues of quasi-isometry. Using similar methodswe also prove that for $n\ge 4$ the braid group $B_n$ is not abstractlycommensurable to any group that splits non-trivially over a "free group-free"subgroup, and the same holds for $n\ge 3$ for the loop braid group $LB_n$. Ourapproach makes heavy use of the Bieri--Neumann--Strebel invariant.

Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups