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