  There are 432 strongly squarefree symmetric bilinear forms of signature
$(2,1)$ defined over $\Z[\sqrt{2}]$ whose integral isometry groups are
generated up to finite index by finitely many reflections. We adapted Allcock's
method (based on Nikulin's) of analysis for the $2$-dimensional Weyl chamber to
the real quadratic setting, and used it to produce a finite list of quadratic
forms which contains all of the ones of interest to us as a sub-list. The
standard method for determining whether a hyperbolic reflection group is
generated up to finite index by reflections is an algorithm of Vinberg.
However, for a large number of our quadratic forms the computation time
required by Vinberg's algorithm was too long. We invented some alternatives,
which we present here.
