Complex sentence: Louis and Hardin separated in 1931, when he had begun a liaison with Alpha Smith, who threatened to sue Armstrong for breach of promise, so he begged Hardin not to grant him a divorce.
Split into simpler sentences: Louis and Hardin separated in 1931, when he had begun a liaison with Alpha Smith. Ms. Smith eventually threatened to sue Armstrong for breach of promise, so he begged Hardin not to grant him a divorce.