A Lie bialgebra is a Lie algebra additionally equipped with a 1-cocycle $\delta: {\mathfrak g}\to \Lambda^2 {\mathfrak g}$ that satisfies the co-Jacobi identity. Non-trivial Lie bialgebras can be obtained from trivial ones (with zero cocommutator $\delta$) by twisting with a classical $r$-matrix. Thus, considering for simplicity the case of an anti-symmetric $r$-matrix $r\in \Lambda^2 {\mathfrak g}$, define $\delta = \partial r$, where $\partial$ is the standard coboundary operator, i.e. $$\delta(x)=[x\otimes 1+1\otimes x,r].$$ It is then easy to check that the cocommutator defined this way is compatible with the Lie algebra structure (i.e. $\delta$ is a 1-cocycle), and the co-Jacobi identity is satisfied in particular when $r$ satisfies the classical Yang-Baxter equation.
My question is whether anyone has ever studied a somewhat analogous but different setup, in which the sign in the above formula for the cocommutator is changed. Thus, I would like to twist by a symmetric "$r$"-matrix $s\in S^2{\mathfrak g}$, and define the cocommutator via $$\delta(x)=[x\otimes 1-1\otimes x,s].$$ The sign is changed as compared to the one in the previous paragraph so that the cocommutator continues to take values in $\Lambda^2 {\mathfrak g}$. It is not hard to check that the cocommutator just defined is no longer compatible with the commutator, there are terms that cancel in the case of the usual sign and do not if the sign is changed. However, I am prepared to accept this (and thus accept that one is no longer in the standard context of Lie bialgebras). Note that I am interested in the case when $s$ is not ${\mathfrak g}$-invariant, the case of a ${\mathfrak g}$-invariant $s$ being somewhat trivial.
So, my question is whether there is still a reasonably nice mathematical structure arising from the above definition, possibly with $s$ satisfying some conditions. In particular, the question is whether there is still any notion of the double, which realises both the commutator and the cocommutator as parts of a commutator of a bigger Lie algebra.
