Letgbe a complex simple Lie algebra with root system ▵. We prove that any classical doubleD (g)is graded by ▵.As a consequence of this fact we obtain thatD (g)≅g⊗A, whereAis a unital com-muative associative algebra of dimension 2. Therefore we have two possibilities forAnilpotent and semisimple. The first case leads to solutions of CYBE and the second case leads to solutions of mCYBE. We obtain an explicit description of Lie bialgebra structures ongin both cases.