Let k: E*E ! [0;1) be a non-negative measurable function on some locally compact separable metric space E. We provide some simple conditions such that the quadratic form with jump kernel k becomes a regular lower bounded (nonlocal, non-symmetric) semi-Dirichlet form. If E = Rⁿ we identify the generator of the semi-Dirichlet form and its (formal) adjoint. In particular, we obtain a closed expression of the adjoint of the stable-like generator in the sense of Bass. Our results complement a recent paper by Fukushima and Uemura [3] and establish the relation of these results with the symmetric principal value (SPV) approach due to Zhi-ming Ma and co-authors [5].