A coboundary Lie bialgebra is a Lie algebra $g$ equipped with a map
$\delta : v \in g \to [v, r]_S \in \Lambda^2 g$, where $[\cdot,\cdot]_S$
is the algebraic Schouten bracket
on the Grassmann algebra $\Lambda g$ and $r \in\Lambda^2 g$ is a solution
of the modified
classical Yang-Baxter equation (MCYBE), i.e. $[v, [r, r]_S ]_S = 0$ for any
$v \in g$. The classification and properties of solutions of the MCYBE are
well-studied mostly for semisimple Lie algebras or when $\dim g \le 3$. To
tackle non-semisimple and higher-dimensional cases, one needs new tools.
In this talk, I will discuss the use of gradations on $g$ and $\Lambda g$
solutions and studying the structure of theMCYBE. Several examples will
be presented to illustrate this approach.