The University of Iowa also got expelled in May 1929 (I think it was largely suspected that they were compensating players and not adhering to rules but the University of Iowa president claimed to not know the reason and the conference never formally announced why). The conference voted to reinstate them after the 1929 season, and they were formally readmitted in February of 1930.I'm no expert on B1G history, cursory research didn't bring anything up except the University of Chicago, which chose to leave the conference in the 1940s after it sacked its football program.
**EDIT** Just found this. Michigan was booted in 1907 for refusing to adhere to league rules limiting it to 5 games a season and 3 years player eligibility. They rejoined in 1916.
Coincidentally, they had to boot like 30 players in December of 1929 by advice of legal counsel due to concerns of compensation (probably to ensure they’d be reaffirmed to the conference without issue).