An integral part of the CPA method to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems is the generation of a suitable triangulation. Recently, the CPA method was revised by using more advanced triangulations and it was proved that it can compute a CPA Lyapunov function for any nonlinear system possessing an exponentially stable equilibrium. This more advanced triangulation scheme includes a simplicial fan close to the equilibrium of the system, for which the Lyapunov function is computed. In this paper we prove a result that allows for a simpler, more general and more efficient generation of this simplicial fan and thus improves the implementation of the CPA method. Moreover, the simplicial fan subdivision at the origin is also of great importance in methods relying on a conic decomposition of the state-space.