We prove that the filtration on the pure braid group on g strands, induced by the lower central series of the Torelli group of a genus g surface with one boundary component, coincides with its lower central series, shifted by one.
In particular, the cubic Jacobi relations in the pure braid group are quadratic relations in the Torelli group.
Sylvain Gervais. Nathan Habegger. "The topological IHX relation, pure braids, and the Torelli group." Duke Math. J. 112 (2) 265 - 280, 1 April 2002. https://doi.org/10.1215/S0012-9074-02-11222-8