We study the anisotropic nature of the Kerr nonlinear response in a beta-barium borate (β-BaB2O4, BBO) nonlinear crystal. The focus is on determining the relevant χ(3) cubic tensor components that affect interaction of type I cascaded second-harmonic generation. Various experiments in the literature are analyzed and we correct the data from some of the experiments for contributions from cascading as well as for updated material parameters. We also perform an additional experimental measurement of the Kerr nonlinear tensor component responsible for self-phase modulation in cascading, and we show that the average value of 14 different measurements is considerably larger than what has been used to date. Our own measurements are consistent with this average value. We also treat data measurements for mixtures of tensor components, and by disentangling them we present for the first time a complete list that we propose as reference of the four major cubic tensor components in BBO. We finally discuss the impact of using the cubic anisotropic response in
ultrafast cascading experiments in BBO.