We revisit the block-Jacobi iterative solver and preconditioner for Hermitian positive definite systems of equations. In order to force convergence of the iterative scheme, the diagonal blocks are nested hierarchically in a binary fashion. The resulting parallel technique, the hierarchical binary Jacobi (hbJ), fits within the nested matrix splitting framework. The convergence criteria of the hbJ are provided and its computational complexity is studied. Comparison theorems based on the levels of the nested iteration are given. A computational study for its use as iterative solver and preconditioner is performed within the class of Wishart random generated Hermitian positive definite matrices. Counterexamples are provided for a selection of desired comparison results.