Efficient analysis and simulation of multiscale systems of chemical kinetics is an ongoing area for research, and is the source of many theoretical and compu- tational challenges. In this paper, we present a significant improvement to the constrained approach, which allows us to compute the effective generator of the slow variables, without the need for expensive stochastic simulations. This is done through finding the null space of the generator of the constrained system. For complex systems where this is not possible, the constrained approach can then be applied in turn to the constrained system in a nested manner, mean- ing that the problem can be broken down into solving many small eigenvalue problems. Moreover, this methodology does not rely on the quasi steady-state assumption, meaning that the effective dynamics that are approximated are highly accurate, and in the case of systems with only monomolecular reactions, are exact. We will demonstrate this with some numerics, and also use the effective generators to sample paths which are conditioned on their endpoints.