  We show that the non-arithmetic lattices in PO(n,1) of Belolipetsky and
Thomson (2011), obtained as fundamental groups of closed hyperbolic manifolds
with short systole, are quasi-arithmetic in the sense of Vinberg, and, by
contrast, the well-known non-arithmetic lattices of Gromov and
Piatetski-Shapiro are not quasi-arithmetic. A corollary of this is that there
are, for all $n\geq 2$, non-arithmetic lattices in PO(n,1) that are not
commensurable with the Gromov--Piatetski-Shapiro lattices.
