Final answer:
Deep to the arbor vitae in the cerebellum, the regions of gray matter are known as the deep cerebellar nuclei, which include the dentate, emboliform, globose, and fastigial nuclei and are crucial for processing cerebellar information.
Step-by-step explanation:
The regions of gray matter deep to the arbor vitae in the cerebellum are referred to as the deep cerebellar nuclei. These include the dentate, emboliform, globose, and fastigial nuclei. They are important for processing information from the cerebellar cortex and sending it to other parts of the brain.
Specifically, the cerebellum can be divided into the spinocerebellum (vermis), vestibulocerebellum (flocculonodular lobe), and cerebrocerebellum (lateral hemispheres), each having distinct functions and connections within the brain.