On some classes of \({\mathbb {Z}}\)-graded Lie algebras

We study finite dimensional almost- and quasi-effective prolongations of nilpotent \({\mathbb {Z}}\)-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi–Malčev and Levi–Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we consider the semisimple case and discuss some examples in which the structural algebras are central extensions of orthogonal Lie algebras and their degree \((-\,1)\) components arise from spin representations.