A lattice \( L \) is called hyperbolic if the associated quadratic form has signature \( (n,1) \). Moreover, the number \( n+1 \) is called the rank of the lattice \( L \). Recall that a vector \( e\in L \) is called a root, if \( k=q(e,e) \) is a positive integer, \( 2q(e,x)\in k\mathbb{Z} \) for all \( x\in L \) (here \( q(\cdot , \cdot) \) denotes the bilinear form induced by the quadratic form) and \( e/m \not\in L \) for all integers \( m>1 \). Each root \( e\in L \) defines an orthogonal reflection \( x\mapsto x-\frac{2q(x,e)}{q(e,e)}e \) in the space \( L\otimes \mathbb{R} \).
Let \( O'(L\otimes \mathbb{R}) \) be the group of motions of the corresponding Lobachevsky space and let \( O(L) \) be the group of automorphisms of the lattice \( L \). Denote by \( O^{(1,2)}(L) \) the subgroup of \( O'(L) =O'(L\otimes \mathbb{R}) \cap O(L) \) generated by all reflections induced by roots \( e\in L \) such that \( q(e,e)\in \{1,2\} \). The lattice \( L \) is called \( (1,2) \)-reflective if \( O^{(1,2)}(L) \) has finite index in \( O'(L) \).
In this paper all \( (1,2) \)-reflective anisotropic hyperbolic lattices of rank \( 4 \) are classified. To obtain the classification, the author modifies the method developed by V. V. Nikulin [On the classification of hyperbolic root systems of rank three. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica (2000; Zbl 1115.17302); Prob. Steklov Inst. Math. 165, 131–155 (1985; Zbl 0577.10019); translation from Tr. Mat. Inst. Steklova 165, 119–142 (1984)] by applying the existing and new results about the geometry of the fundamental polyhedron.