is given by $[M/UM, UM; a, b]$. Conversely, given an (indecomposable) special representation $[V, W ; \psi_1, \psi_2]$ of $Q$, the corresponding (indecomposable) $G$-module $M$ is given by $M = V
\oplus W$ where $a(\alpha, \beta) := (0, \psi_1(\alpha))$ and $b(\alpha, \beta) := (0, \psi_2(\alpha)).$

We will use this translation between the representations of the Klein group and the Kronecker Quiver freely through out the paper.

If $M = [V_1, V_2; a, b]$ is an indecomposable projective-free representation, then we have $$\begin{aligned}
V_1 &= 0 \; \Leftrightarrow \; M = 0 \\
   V_2 &= 0\; \Leftrightarrow \; M = k.\end{aligned}$$ So henceforth it will be assumed that the spaces $V_1$ and $V_2$ are non-zero, i.e., $M$ is an indecomposable projective-free and a non-trivial representation of $V_4$.

We begin with some lemmas that will help streamline the proof of the classification theorem. The proofs of these lemmas will be deferred to the last section. It should be noted that these lemmas are also of independent interest.

[@ben-1]\[lemma1\] Let $M$ be a projective-free $V_4$-module given by $[V_1, V_2; a, b]$. Then we have the following.

1.  $M$ contains a copy of $\Omega^l(k)$ for some positive integer $l$ if and only if the 