  We develop a method to compute certain mutations of quivers with potentials
and use this to construct an explicit family of non-degenerate potentials on
the exceptional quiver $X_7$.
  We confirm a conjecture of Geiss-Labardini-Schroer by presenting a
computer-assisted proof that over a ground field of characteristic 2, the
Jacobian algebra of one member $W_0$ of this family is infinite-dimensional,
whereas that of another member $W_1$ is finite-dimensional, implying that these
potentials are not right equivalent.
  As a consequence, we draw some conclusions on the associated cluster
categories, and in particular obtain a representation theoretic proof that
there are no reddening mutation sequences for the quiver $X_7$.
  We also show that when the characteristic of the ground field differs from 2,
the Jacobian algebras of $W_0$ and $W_1$ are both finite-dimensional. Thus
$W_0$ seems to be the first known non-degenerate potential with the property
that the finite-dimensionality of its Jacobian algebra depends upon the ground
field.
