Why is a dog's saliva impure while its fur is not? And what is the view of the four madhhabs on this, along with the evidence?

The preponderant view is that of the majority, which holds that a dog is impure in all its parts. Therefore, if a garment, body, or place is touched by its saliva or wet body, it must be purified. They based their argument on the hadith: "If a dog licks the vessel of one of you, he should spill it and then wash it seven times," and "The purification of the vessel of one of you, if a dog licks it, is to wash it seven times, the first with earth." This hadith indicates impurity because the word "purification" (ṭahūr) is used for either ritual impurity (ḥadath) or physical impurity (khabath). Since there is no ritual impurity on a vessel, physical impurity is thereby confirmed.