We will examine the ternary differential system with quadratic nonlinearities. For this system comitants, contravariants and mixed comitants, as well Lie algebra of the centroaffine group were constructed in [1]. Using these elements the ternary differential system having 27 terms can be reduced to a ternary differential system with 18 terms in the right-hand sides. The quadratic homogeneities of the system can be represented as a product of two linear factors, one of them being a phase variable and common in each equation. For the obtained system it is important to belong to a centroaffine invariant variety. The obtained system is called the Iacobi ternary differential system. It is easy to see that this system generate the Lorentz system from meteorology [2] and the system of the intrinsic transmission dynamics of tuberculosis in a society [3]. We pay attention to the problem of determining the necessary and sufficient centroaffine invariant conditions which allow to transform the quadratic ternary differential system to the Iacobi ternary differential system. It was proved the following result: Theorem 1. The necessary conditions for the quadratic ternary differential system to be reduced to the Iacobi ternary differential system by a centoaffine transformation are that the discriminant of the linear combination of the quadratic right-hand sides and the resultants of the quadratic homogeneities to vanish. These conditions are centroaffine invariant.