A few words from "The Symbolism of Chess" by Titus Burckhardt:
"The cyclical symbolism of the chess-board was known to King Alphonsus the Wise, the famous troubadour of Castille, who in 1283 composed his Libros de Acedrex, a work which draws largely from Oriental sources. Alphonsus the Wise also describes a very ancient variant of the game of chess, the "game of the four seasons", which takes place between four partners, so that the pieces, placed in the four corners of the chess-board, move in a rotatory direction, analogous to the movement of the sun. The pieces must have the colours green, red, black, and white; they correspond to the four seasons: spring, summer, autumn, and winter; to the four elements: air, fire, earth, and water, and to the four organic "humours". The movement of the four camps symbolizes cyclical transformation. This game, which strangely resembles certain "solar" rites and dances of the Indians of North America, brings into relief the fundamental principle of the chess- board."
"If the significance of the different chessmen is transposed into the spiritual domain, the king becomes the heart, or spirit, and the other pieces the various faculties of the soul. Their movements, moreover, correspond to different ways of realizing the cosmic possibilities represented by the chess-board: there is the axial movement of the "castles" or war chariots, the diagonal movement of the "bishops" or elephants, which follow a single colour, and the complex movement of the knights. The axial movement, which "cuts" through the different "colours", is logical and virile, while the diagonal movement corresponds to an "existential"—and therefore feminine—continuity. The jump of the knights corresponds to intuition."
Some authors agree that the four colours represent the elements: air, fire, earth, and water, but let's see what these elements can represent:
The four major conditions of humanity or four levels of perception
Rituals and Sufi Spiritual Beliefs. Idries Shah
HUMANITY (black colour - Earth)
Abid (a worshipper)
The ordinary state. Being in contact with, and partaking of the nature and behaviour of ordinary humanity.
DISCIPLESHIP (white colour - Water)
Being on the Path. Being in harmony with a Teaching Master.
REAL CAPACITY (green colour - Air)
When progress starts. Being in contact with the Founder of the Teaching.
ATTUNEMENT WITH THE DIVINE (red colour - Fire)
The final condition. Being in harmony with Absolute Truth.