The Onsager Lie algebra O is an infinite-dimensional Lie algebra defined by generators A and B and relations \([A, [A, [A, B]]] = 4[A, B]\) , and \([B, [B, [B, A]]] = 4[B, A]\) . Using an embedding of O into the tetrahedron Lie algebra \(\boxtimes \) , we obtain four direct sum decompositions of the vector space O, each consisting of three summands. As we will show, there is a natural action of \(\mathbb {Z}_2 \times \mathbb {Z}_2\) on these decompositions. For each decomposition, we provide a basis for each summand. Moreover, we describe the Lie bracket action on these bases and show how they are recursively constructed from the generators A and B of O. Finally, we discuss the action of \(\mathbb {Z}_2 \times \mathbb {Z}_2\) on these bases and determine some transition matrices among the bases.
