From the Tits Form to A₂ × A₁: A Signed-Edge Bilinear Form on a Four-Vertex Quiver

For the four-vertex six-arrow strongly connected quiver Q with vertex set {L, S, T, C}, the cycle space H₁(Q; ℝ) is three-dimensional, generated by the four-gon γ₁, the shorter cycle γ₂, and the digon γ₃. The canonical Tits form of Q on this basis is indefinite, with diagonal (−4, −2, 0) and one isotropic generator, and realizes no Cartan matrix of Dynkin type. Taking a single structural input beyond Q, a partition of the edges into signed and unsigned with E_signed = {a₃}, the sole exit of the in-degree-3 vertex T, this paper constructs a modified symmetric bilinear form B whose matrix on {γ₁, γ₂, γ₃} is the Cartan matrix of A₂ × A₁. B is shown unique up to diagonal scaling once the partition is fixed, and two constructions of B, one that derives the partition from Q's structure and one that assumes it as a premise, are shown to produce the same form, both importing the partition and differing only in its source. The reflections in B generate the Weyl group S₃ × ℤ/2, and the Cartan-Killing correspondence returns the Lie algebra su(3) ⊕ su(2). A rank-4 extension B̃, adjoining a temporal generator γ_τ with B̃(γ_τ, γ_τ) = 4, returns su(3) ⊕ su(2) ⊕ u(1), with structural invariant k₁ = 2 and det(B̃) = 24 = 4!. The construction is stated as mathematics; the source and justification of the signed/unsigned partition are supplied by the companion papers. **Keywords:** quiver, path algebra, cycle space, Betti number, Tits form, bilinear form, Cartan matrix, A₂ × A₁, Weyl group, Cartan-Killing correspondence, su(3) ⊕ su(2), su(3) ⊕ su(2) ⊕ u(1), determinant