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

The classification of Rank 3 Reflective Hyperbolic Lattices over Z[\sqrt{2}]