Reducible $p$-representations of a simple three-dimensional Lie $p$-algebra

We study the category $\mathscr{P}$ of restricted finite-dimensional representations of the simple $3$-dimensional Lie algebra $L$. We show that $\mathscr{P}$ is equivalent to the sum of some categories of diagrams over finite dimensional vector spaces. We find the indecomposable objects in the latter categories. Thus we obtain a classification of indecomposable restricted representations of $L$ and an effective method of their construction.