In a recent series of papers, Gill and co-workers have further developed techniques for directly factorizing the Coulomb operator and its long-ranged Ewald counterpart. Such factorizations turn the twoparticle operator into sums of products of one-particle terms, for which highly efficient implementations are possible. In this work, we show that the same factorization may be derived by an analytic spherical integration. From that point, we demonstrate two significant improvements: cubature to perform the integration with arbitrary precision, and an adaptive scheme for determining the Ewald separation parameter, α. The introduction of cubature results in a highly compact functional form, involving the evaluation of simple trigonometric quantities. The adaptive α modification extends the applicability of the factorization to any distance domain, resulting in a O(N log(N )) scaling algorithm when evaluated using a treecode-like approach. Because the distinguishing features of this novel method are the Spherical Grid and Treecode components, we hereafter refer to it as SGT for brevity. The following section will describe the formulation of SGT for electrostatics, with Appendix A providing details of how to generalize the method to other kernels.