The forms and representations of the Lie algebra $sl_2(\mathbb Z)$

We study the structure of integral $p$-adic forms of the splitting three-dimensional simple Lie algebra over the field of $p$-adic numbers. We discuss the questions of diagonalizability of such forms and description for maximal diagonal ideals. We consider torsion-free finite-dimensional modules over the splitting three-dimensional simple Lie algebra with integral and $p$-adic integral coefficients. We describe diagonal modules, demonstrate finiteness of the number of modules in each dimension, and prove a local-global principle for irreducible modules.