The red nucleus is not a nuclei of the thalamus.
What is the thalamus?
Thalamus is a part of the brain that occupies most of the diencephalon. It is an egg-shaped, paired mass. It has many nuclei, each of which has a different role. The roles include relaying information to regulating alertness and consciousness.
The anatomical core of the brain, dorsal part of the diencephalon is where the thalamus is located. Adjacent to the hypothalamus, the thalamus is situated deep in the brain in the cerebral cortex.
Each side of the thalamus has more than 20 nuclei, which also consists of the gray matter. The nuclei are, at times, referred to as thalamic bodies. They are present in groups that are distinguished by layers of white matter that is called laminae.