 Interval representations of an equioriented three-vertex quiver

For $1\to 2\to 3$, $I[a,b]$ has $k$ on the interval of vertices $a,\dots ,b$, zero elsewhere and identity maps along its internal arrows. These six representations are exactly the indecomposables. To decompose $V_1\stackrel{f}{\to }{V}_2\stackrel{g}{\to }{V}_3$, split $\mathrm{im}f\cap \ker g$, its complements inside each of $\mathrm{im}f$ and $\ker g$, and a complement to their sum. Lift the image basis through $f$ and transport the complementary nonkernel basis through $g$. This yields interval blocks directly, without the Gabriel theorem.