Socially correct behavior requires constant observation of the social environment. Behavior that was appropriate a few seconds ago is not guaranteed to be appropriate now. The brain keeps the eyes focused on the current social space and constantly updates its internal representation of the environment and social context. Monitoring the behavior of others is essential for this updating. The neural systems involved in perceiving the actions of others have been explored extensively, but the detailed, quantitative character of the system at the single-cell level remains poorly understood. To address this question, we used the new technique of multidimensional recording to record neuronal activity in monkeys simultaneously from ventral premotor cortex (PM) and parietal cortex in the left hemisphere while they performed a food grab task. Motion-related (MR) response was shown by 35% (52/148) of PM neurons and 54% (94/174) of parietal neurons, meaning their activity increased in response to various combinations of arm motions made by self and/or other. Both areas showed robust lateralized preference to Self-Right action. When it came to recognizing the actions of the other monkey, PM-MR neurons showed the same kind of right-arm preference as self-action while parietal-MR neurons, in contrast, did not show arm preference. And while both areas discriminated self-action from other, a significantly larger proportion of PM-MR neurons did so. These results suggest that PM neurons provide information about an action's agent and effector as primitives of action cognition within the mirror neuron network,while parietal neurons represent social space and participate in the recognition of another agent's actions in relation to one's own actions within the parieto-prefrontal network.