To advance the space-time Energetic Boundary Element Method (EBEM) in 3D elastodynamics [1], we study the characteristic singularity of the double-layer operator, which is involved in the direct boundary integral formulation. On the basis of a decomposition of the time-dependent point load traction Green’s function, we employ a regularized Boundary Integral Equation (BIE) and we discretize it by means of a Galerkin-type EBEM (introduced for the first time in [2]) with double analytical integration in the time variable. However, one of the main difficulties of this approach is the efficient approximation of remaining weakly singular double space integrals, whose accurate computation is a key issue for the stability of the method. In particular, the integration domains are generally delimited by the wave fronts of the primary and the secondary waves. By analyzing the geometric characteristic of these domains, we develop an ad-hoc quadrature strategy, where the outer integrals are computed efficiently by the Gaussian quadrature (with a small number of points), while the inner integrals are evaluated with respect to polar coordinates and expressed by analytical formulations. The effectiveness of the proposed approach is illustrated via two benchmark problems. [1] A. Aimi, S. Dallospedale, L. Desiderio, C. Guardasoni. A space-time Energetic BIE method for 3D Elastodynamics. The Dirichlet case. Computational Mechanics, 72(5), 2023, pp. 885–905. [2] A.~Aimi, M.~Diligenti. A new space-time energetic formulation for wave propagation analysis in layered media by BEMs. International Journal for Numerical Methods in Engineering \textbf{75}(9), (2008), 1102--1132.