The article introduces new bases for analyzing game spaces and the Shapley value. A new game called the (T,k)-intermediate game is created, where 1 is assigned to a coalition with k players in a fixed coalition T. These new bases maintain important properties of commander games. When games are expressed as a combination of these bases, the coefficients for single players match the Shapley value. The bases also reveal the null space of the Shapley value. By applying these bases, the relationship between the Shapley value and the prenucleolus can be analyzed, shedding light on the mathematical structure of their coincidence region.