Picking $l$ to be minimal, we can conclude from lemma \[lemma2\] that $\Omega^l(k)$ is a direct summand of $M$. Since $M$ is indecomposable, we have $M \cong \Omega^l(k)$. This gives the modules in item 3(b).\
*Case 2 :* $\dim V_1 < \dim V_2$. Dualising $M = (V_1, V_2;
a , b)$, we get the dual representation $M^* = (V_2^*, V_1^*; a^*,
b^*)$ which is also indecomposable. Now $\dim V_2^* > \dim V_1^*$, so by Case(1) we know that $M^* \cong \Omega^l(k)$ for some $l$ positive. Taking duals on both sides and invoking lemma $\ref{lemma3}$, we get $M \cong \Omega^{-l}(k)$. This recovers the modules in item 3(c).

This completes the proof of the classification of the indecomposable representations of $V_4$.

Some applications
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Having a good classification of the indecomposable representations of a finite group helps a great deal in answering general module theoretic questions. In this section, we illustrate this by proving some facts about module over the Klein group. Note that we don’t know of any direct proofs of the statements below that do not depend on the classification of the indecomposable representations.

Heller Shifts of the $V_4$-representations.
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We will show how our knowledge of the representations of 