  The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which
is equivalent to the isotropic system introduced by Bouchet. In this paper we
discuss four notions of connectivity related to isotropic matroids and
isotropic systems. We show that the isotropic system connectivity defined by
Bouchet is equivalent to vertical connectivity of $M[IAS(G)]$, and if $G$ has
at least four vertices, then $M[IAS(G)]$ is vertically 5-connected if and only
if $G$ is prime (in the sense of Cunningham's split decomposition). We also
show that $M[IAS(G)]$ is $3$-connected if and only if $G$ is connected and has
neither a pendant vertex nor a pair of twin vertices. Our most interesting
theorem is that if $G$ has $n\geq7$ vertices then $M[IAS(G)]$ is not vertically
$n$-connected. This abstract-seeming result is equivalent to the more concrete
assertion that $G$ is locally equivalent to a graph with a vertex of degree
$<\frac{n-1}{2}$.
