The Tetrahedron algebra, the Onsager algebra, and the loop algebra

Let denote a field with characteristic 0 and let T denote an indeterminate. We give a presentation for the three-point loop algebra via generators and relations. This presentation displays S4-symmetry. Using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras, each of which is isomorphic to the Onsager algebra.