In a previous work, we have presented a new formulation of the uniform version of the Fast Multipole Method (FMM) for the Laplace equation by using matrix products that can be efficiently computed thanks to the BLAS (Basic Linear Algebra Subprograms) routines. We propose here to extend this formulation to the adaptive version of the FMM: this requires the conception of a new data structure for the octree, namely the octree with indirections, which is efficient for both uniform and non-uniform distributions, as well as a detection mechanism of the available uniform areas in non-uniform distributions. In comparison with other M2L computation schemes (block FFT, rotations and plane wave expansions) in the case of non-uniform distributions of particles, our BLAS version appears to be the fastest for the common precisions used when one solves the Laplace equation.