Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties:
(1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$.
(2) $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] \in g \otimes g \otimes g$ is $g$-invariant.
Here the $g$-action on $g \otimes g$ and on $g \otimes g \otimes g$ is by the adjoint representation on each factor.
The element $r$ can be written as a four by four matrix. Has this $r$ (all possible $r$ which satisfies conditions (1) and (2)) been computed? Any help will be greatly appreciated!
