Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank 4

A hyperbolic lattice is said to be $(1{,}{\kern1pt}2)$-reflective if its automorphism group is generated by $1$- and $2$-reflections up to finite index. We prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic cocompact reflection group in three-dimensional Lobachevsky space contains an edge with sufficiently small distance between its framing faces. Using this fact, we obtain a classification of $(1{,}{\kern1pt}2)$-reflective anisotropic hyperbolic lattices of rank $4$.