This paper deals with solving \(Au=f\), where \(A\) is a symmetric and positive definite matrix, with two-level hierarchical basis (TL) or two-grid (TG) iterative methods. Fundamentally, the methods differ in the two-component decomposition of the underlying vector space of unknowns: in the TL method, the sum is direct, which is not necessarily true for the TG method. The authors construct \(B_{\text{TL}}\), a TL preconditioner, and infer the best value of the constant that appears in the spectral equivalence relation between \(A\) and \(B_{\text{TL}}\). An analogous analysis for the TB method results in the corresponding preconditioner and spectral equivalence constant. The constants determine the convergence factors of the TL and TG preconditioners. Three mutually equivalent necessary conditions for uniform convergence of the TG method are given. The final part of the paper focuses on constructing an efficient window-based algebraic two-grid method.