Fuzzy implications play an important role in both approximating reasoning and fuzzy control, and there are many generation methods for fuzzy implications for various applications. The extended threshold generation method for fuzzy implications can induce some implications differ from the usual implications, such as f-, g-implications, R-implications, S-implications, and those derived from the ordinal sum method. In this paper, we focus on the investigations of the preservation of some important properties under minimal necessary conditions similar to the e-threshold generation method, such as the neutral property, the exchange law, the weak importation law, the contraction law, and the distributivity property over t-norms and t-conorms.