The adaptive explicit-implicit tau-leaping method with automatic tau selection [Y. Cao, D. T. Gillespie and L. R. Petzold, J. Chem. Phys., 126(22):224101 (2007)] is a flexible algorithm for accelerated stochastic simulation of chemically reacting systems. It combines the advantages of different simulation schemes and is particularly useful when a system changes its dynamical behavior over time in the sense that it behaves well in some time periods but possesses stiffness in other time periods. However, the ingredients necessary to fully understand and implement the algorithm are spread over several papers, not always consistent in terminology and notation, which considerably hampers and possibly even prevents accessibility and widespread practical use. We present a streamlined description of the algorithm using a unified terminology and notation and introduce significantly simplified versions of two major ingredients, namely the step size selection and the switching mechanism between the sub-algorithms. This should be greatly helpful for researchers not yet familiar with but interested in tau-leaping methods as well as for practitioners in actually implementing and applying the method.