indecomposable representation given by $$\xymatrix@=2em{
        &  \bullet \ar[dr]_b & & \bullet\ar[dl]^a \ar[dr] &  & \bullet \ar[dl]  & \bullet  \hspace{4 mm}
          \bullet \hspace{4 mm} \bullet & \bullet \ar[dr]
          & & \bullet \ar[dl] \\
        &   & \bullet & & \bullet &  & &  & \bullet &
        }$$

2.  If $n$ is a negative integer, then $\Omega^n (k)$ is a $2n+1$ dimensional indecomposable representation given by $$\xymatrix@=2em{
    & & \bullet \ar[dr]^b \ar[dl]_a & & \bullet \ar[dl] \ar[dr] & &
          & & \bullet \ar[dl] \ar[dr] &  \\
     &    \bullet &  & \bullet & & \bullet & \bullet  \hspace{4 mm}
          \bullet \hspace{4 mm} \bullet  &  \bullet & & \bullet
    }$$

We now give the proof of the classification theorem assuming these lemmas. The lemmas will be proved in the last section. Let $M = (V_1,
V_2; a, b)$ be an indecomposable projective-free representation of $V_4$. We will show that $M$ is isomorphic to one of the representation that appear in items (2) it is even dimensional, and to those in item (3) if it odd dimensional.

Even dimensional representations
--------------------------------

Let $M$ be an even dimensional ($2n$ say) indecomposable representation. We break the argument into cases for clarity.\
*Case 1:* $\det (a + \lambda b)$ 