We consider an algebraic multilevel preconditioning method for SPD matrices resulting from finite element discretization of elliptic PDEs. The method is based on element agglomeration, and, in particular, designed for non-M matrices. Granted that the element matrices at the fine-grid level are given, we further assume that we have access to some algorithm that performs a reasonable agglomeration of fine-grid elements at any given level. The coarse-grid element matrices are simply Schur complements computed from the locally assembled fine-grid element matrices, i.e., agglomerate matrices. Hence, these matrices can be assembled to a global approximate Schur complement. The elimination of fine-degrees of freedom in the agglomerate matrices is done without neglecting any fill-in. This offers the opportunity to construct a new kind of incomplete LU factorization of the pivot matrix at every level, which is done by means of a slightly modified assembling process. Based on these components an algebraic multilevel preconditioner can be defined for more general SPD matrices. The method can also be applied to systems of PDEs. A numerical analysis shows its efficiency and robustness.