The Four-Gon Quiver, Whose Bilinear Invariants Produce A₂ × A₁

The four-gon quiver Q, the directed graph with four vertices and six directed edges, has first Betti number three, one vertex of in-degree three, and three independent cycles of lengths four, three, and two. The path algebra kQ of Q is constructed in the standard sense (Assem et al., 2006). The quotient algebra A = kQ/I modulo the binary cycle relations is infinite-dimensional. The representation category Rep(A, κ) under the convergence constraint κ at the in-degree-three vertex is finite-type with three indecomposable representations. The canonical Tits form of Q is indefinite with one isotropic generator and does not select a Dynkin Cartan matrix. A second symmetric bilinear form B, constructed from a signed-edge partition of the convergence vertex's exit, has the Cartan matrix of A₂ × A₁ as its matrix in the cycle-generator basis. The Cartan-Killing correspondence on A₂ × A₁ returns the semisimple Lie algebra su(3) ⊕ su(2). Five connections to active research programs are identified, quiver varieties (Nakajima, 1994), D-brane quivers (Douglas & Moore, 1996), Hall algebras and quantum groups (Ringel, 1990), cluster algebras (Fomin & Zelevinsky, 2002), and categorification (Khovanov, 2000). The graded dimensions of kQ at lengths 0, 1, 2, 3, 4 are 4, 6, 8, 12, 18. The number of depth-n iterated four-gon (Hamiltonian cycle) traversals is 4ⁿ. The constructions are checkable by any quiver representation theorist using standard methods. Q is shown to be of infinite mutation type, so it is not mutation-equivalent to any Dynkin quiver, the witness being a three-step mutation sequence producing an arrow of multiplicity three. The exchange matrix is blind at the seed to the digon, whose two arrows cancel in it, and the blindness is exact: deleting both digon arrows leaves the exchange matrix unchanged, so Q and the four-arrow quiver without the digon share every cluster-algebraic invariant while differing in first Betti number, three against one. This is why the mutation-theoretic and bilinear-form classifications of Q disagree. Four predictions are stated, including that Ringel's Hall algebra construction applied to kQ/I is conjectured to yield quantum group structure. **Keywords:** path algebra, quiver, finite representation type, indecomposable representations, Cartan matrix, root system, A₂ × A₁, Cartan-Killing correspondence, semisimple Lie algebra, su(3) ⊕ su(2), Tits form, Euler form, symmetric bilinear form, Hall algebra, cluster algebra