The perturbative SO(3) invariant of rational homology 3-spheres recovers from the universal perturbative invariant

For a Lie algebra g and its representation R, the quantum (g, R) invariant of knots recovers from the Kontsevich invariant through the weight system derived from substitution of g and R into chord diagrams. We expect a similar property for invariants of 3-manifolds; for a Lie group G, the perturbative G invariant of 3-manifolds should recover from the universal perturbative invariant defined in [25] through the weight system derived from substitution of the Lie algebra of G. In this paper we give a rigorous proof of the recovery for G=SO(3).