Journal of Lie Theory

\def\g{{\frak g}} We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of $C$-admissible pair as introduced by H. Rubenthaler and J. Nervi for semi-simple and affine Lie algebras. If $\g$ is a Kac-Moody Lie algebra (with Dynkin diagram indexed by $I$) and $(I,J)$ is such a $C$-admissible pair, we construct a $C$-admissible subalgebra $\g^J$, which is a Kac-Moody Lie algebra of the same type as $\g$, and whose root system $\Sigma$ grades finitely the Lie algebra $\g$. For an admissible quotient $\rho: I\to\overline I$ we build also a Kac-Moody subalgebra $\g^\rho$ which grades finitely the Lie algebra $\g$. If $\g$ is affine or hyperbolic, we prove that the classification of the gradations of $\g$ is equivalent to those of the $C$-admissible pairs and of the admissible quotients. For general Kac-Moody Lie algebras of indefinite type, the situation may be more complicated; it is (less precisely) described by the concept of generalized $C$-admissible pairs.