Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra, and let $r\in\bigwedge^2\mathfrak{g}$ be a solution of the Yang-Baxter equation. The Yang–Baxter equation states that for all $x\in\mathfrak{g}$, $$\operatorname{ad}_x[r,r]=0,$$ where $[r,r]$ represents the Schouten–Nijenhuis bracket and the adjoint action $\operatorname{ad}_x$ is extended to $\bigwedge^\ast\mathfrak{g}$ as a derivation.
Now, consider a linear map $\delta:\mathfrak{g}\to\bigwedge^2\mathfrak{g}$, which is a 1-cocycle, meaning it satisfies the condition $$\delta([x,y])=\operatorname{ad}_x\delta(y)-\operatorname{ad}_y\delta(x).$$
I want to determine the conditions under which the 1-cocycle $\xi=\delta+\operatorname{ad} r$ defines a Lie bialgebra, i.e., the transpose map $\xi^t:\bigwedge^2\mathfrak{g}^\ast\to\mathfrak{g}^\ast$ defines a Lie bracket on the dual vector space $\mathfrak{g}^\ast$. In particular, is it necessary for $(\mathfrak{g}, \delta)$ to be a Lie bialgebra?
A particular case I know: if $\delta = \operatorname{ad} R$ is a 1-coboundary with $[R,R]$ ad-invariant, the condition is that the two Poisson tensors associated to $r$ and $R$ have to be compatible, i.e., their Schouten bracket is zero.
\operatorname{ad}in place of\mathrm{ad}(and similarly for other operators) spaces better: compare, e.g., $\operatorname{ad} R$\operatorname{ad} R, where spacing is handled automatically, with $\mathrm{ad}\,R$\mathrm{ad}\,R, where it must be added manually. I have edited accordingly. $\endgroup$