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Two techniques used to show a matrix pattern is spectrally arbitrary are the nilpotent-Jacobian method and more recently the nilpotent-centralizer method. This paper presents generalizations of both techniques, which are then used to show that certain non-spectrally-arbitrary patterns are inertially arbitrary. A flaw in a method used in three previous publications of inertially arbitrary patterns is discussed. By using the techniques developed here, it is shown that all of the patterns in the three papers affected by the flaw are neverthless inertially arbitrary.