This is Dakshninamurthy. Sitting in Sukhasana Siva is espousing wisdom. In Siva temples, Dakshinamurthy is kept in a niche on the outer wall (Kosta Murthy) facing South. Dakshin means south.
In South Indian Images of Gods and Goddess, H K Sastry links Dakshinamurthy to Veerabhadra/Sati/Daksha episode.
This wooden carving shows Dakshninamurthy sitting on a hill, with his right leg resting on Apasmara. He is flanked by two sages. The identification of the sages is not practical and it could be any of the eight sages. Dakshinamurthy is always shown without his consort.
Dakshninamurthy’s attribute includes Akshamala, Serpent, waterpot (?) and Pustaka. The third eye is prominent. His lower right hand is in cinmudra or Janana mudra. Note the matted hair and tree above his head. Note the right and left earrings are different and it is a typical feature of Siva.
This site has some examples of bronze Vigrahas of Dakshinamurthy.
This was probably a part of a temple chariot. The sculpture is about 30 cm in height and carved out of a single block of wood. The thickness is about 7 cm and the deepest part of the carving is about 3 cm. Time does take its toll on wooden pieces.
The sculpture is from Tamil Nadu and may not be older than the nineteenth century.