Three-subspace quiver ($S_3$)

The subspace quiver with three sources reduces to the problem of three subspaces in one vector space. The three-subspace decomposition gives eleven indecomposables with vertex dimensions at most one, and one exceptional indecomposable of the three-subspace quiver.

Every triple of subspaces is a direct sum of one-dimensional membership blocks and two-dimensional three-line blocks. Split off the triple intersection, then the pairwise intersections using projections killing the third subspace. Next split complements to $U_i\cap (U_j+U_{\ell })$, using projections killing the other two subspaces. The remaining subspaces are pairwise disjoint and each lies in the sum of the other two. Their sum is $U_1\oplus U_2$, with $U_3$ the graph of an isomorphism $U_1\to U_2$. A graph basis splits this into two-dimensional blocks. A complement to the total span supplies zero-membership blocks. This elementary argument classifies the three-subspace quiver.

Take sources $k,k,k$, sink $k^2$, and arrows sending $1$ to $e_1,e_2,e_1+e_2$. An endomorphism preserving these three lines must be scalar, so the representation has endomorphism ring $k$ and is a brick module. It is the only indecomposable of the three-subspace quiver with a two-dimensional sink.