Four bases for the Onsager Lie algebra related by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ action

The Onsager Lie algebra $O$ is an infinite-dimensional Lie algebra defined by generators $A$, $B$ 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 $A$, $B$ 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.