We generalise classical estimates by Möller’s (Numerische Integration: Tagung im Mathematischen Forschungsinstitut Oberwolfach, pp 221–230, 1979, [22]) for the minimal number of nodes in cubature formulas for symmetric integrals to obtain a lower bound on the number of paths in the support of cubature measures on Wiener space. Motivated by this analysis, we construct a family of degree-5 cubature measures on Lie polynomials that attain this lower bound when implemented with corresponding Gaussian cubature formulas of minimal support. Our construction yields measures with significantly smaller support than the formulas introduced by Lyons and Victoir (Proc Royal Soc Lond. Ser A: Math Phys Eng Sci 460(2041):169–198, 2004, [20]) in both low- and high-dimensional noise settings.