The problem of maximality of arithmetic subgroups in orthogonal groups of the type $B_n$

We obtain an explicit classification of the maximal arithmetic subgroups of decomposable orthogonal groups of type $B_n$ over the field of rational numbers or the field of rational functions of a single variable with a finite field of constants (of characteristic not equal to 2). In passing, we solve the problem concerning the classification of maximal compact subgroups of decomposable orthogonal groups of type $B_n$ over $Q_2$.