In differential geometry and (pseudo-)Riemannian manifolds, Riemman metric tensor plays a significant role in deducing basic formulas and equations. In previous work, we developed a general computational theory for indexed polynomials involving Riemman metric tensor and its differential forms by extending Gr\"obner basis theory. However, the general theory for finding the canonical form of an indexed Riemman metric tensor monomial has a factorial complexity. This work puts forward a much more efficient method with polynomial complexity. First, we study the invariance of Leibniz expansion of indexed Riemann metric tensor monomials under differential operators and monoterm symmetries. Then, simpler generators of the ideal generated by the basic syzygies are explored. Finally, we find a Gr\"obner basis of the generators, by investigating the properties of the leading terms of S-polynomials, and prove that the reductions with respect to the Gr\"obner basis has polynomial complexity.