Many sources will state that the first three stomachs of cows are used in tripe. The only reasoning I have found for not using one of the stomachs is listed by sciencedirect.com:
The third compartment of the ruminant stomach is the omasum. The internal wall is in the form of deep, thin folds like the pages of a book. This appearance accounts for the popular name for the omasum, which is ‘bible.’ Although the omasum has a delicate flavor and texture, it is not commonly used as tripe because of the difficulty of cleaning the stomach contents from between the folds. The fourth part of the ruminant stomach is the abomasum, sometimes called the reed.
Is the logic the same as often not using the third stomach (omasum) but to a larger degree? Is the abomasum just too difficult to clean for use as tripe?