<p>In this paper, first, the Kellogg-type iterative method is presented for solving linear system of equations using arbitrary pairs of splittings. As part of the analyses, a sufficient condition is provided under which both HSS method and its corresponding Kellogg-type iteration methods exhibit nearly identical convergence speed. Furthermore, we develop a new class of matrix splittings and their induced preconditioners, specifically designed for the underlying block systems in the Kellogg-type iterations. Under certain conditions, we prove that the proposed preconditioners yield a positive definite preconditioned matrix, improving the convergence speed of Krylov subspace methods. Some experimental results are included to numerically confirm the theoretical discussions and illustrate the effectiveness of proposed approaches.</p>