Indecomposable objects and triangles in bounded derived category $D^{b}(\operatorname{mod} A)$ of finite dimensional algebra A

Question

Let $A$ be a finite dimensional algebra given by the quiver $$ \begin{align} \swarrow\searrow\\ \searrow\swarrow \end{align} $$ with the relation $\beta\alpha-\delta\gamma$.

Let's consider the bounded derived category $D^{b}(\operatorname{mod} A)$,
Question: the indecomposable objects and triangles are ...? Can it be shown as Auslander-Reiten quiver? (see $AR$-quiver for hereditary algebras in reference [1])

Thank you!

Reference

[1] Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119. Cambridge University Press, Cambridge, 1988. pp. x+208.

Answer 1

Your quiver algebra is a tilted algebra of hereditary Dynkin type $D_4$ and thus derived equivalent to $D_4$ and thus all is known including the AR-quiver. This example is also treated (with more details and references on the general situation in the article) as example 2.9 in the article "Derived categories and tilting" by Bernhard Keller in the Handbook of tilting theory.