Starting from the minimal complex matrix algebra M2(C), this paper shows that two biquaternion products suffice to derive the complete conservation structure of a relativistic perfect fluid and its dilatation current, without postulating a Lagrangian and without invoking Noether's theorem as an external tool.The product M = R^T G, formed from the spacetime position four-vector R and the energy-momentum four-vector G, decomposes automatically into the action density, angular momentum density, and moment-of-energy density as algebraically forced channel outputs. The closure condition dM = 0 packages all three conservation laws into a single Maurer-Cartan flatness condition on a Lie-algebra-valued current in the adjoint representation of the Lorentz group, a structure that follows from the asymmetric transformation law of the biquaternion transpose and requires no additional input.The perfect-fluid reduction projects M onto the adjoint orbit of the fluid velocity, yielding the relativistic Lagrangian fluid equations in comoving coordinates whose gamma-scaling encodes the dilatation structure of the flow. A companion product J_D = R(U^T G) constructs the dilatation current explicitly: under the perfect-fluid constraint it reduces to J_D = eR, a pure Lorentz four-vector whose conservation is the relativistic virial theorem and whose sole surviving constraint in the thin-disk limit is the Euler homogeneity relation div(r) + 1 = 0 — the differential signature of a renormalisation-group fixed point, derived here without assuming scale invariance.The construction demonstrates that the Lagrangian, the Euler-Lagrange equations, and the Noether currents are outputs of the algebraic structure of M2(C) rather than its foundation, and identifies the biquaternion transposed product as the minimal algebraic operation from which the standard framework's results follow as necessary consequences.