Though the father-son relationship between Troy and Cory is the most important one in the play, it's also important to examine the relationship between Lyons and Troy.
Lyons is the product of Troy's first marriage. He is thirty-four years old. When he is introduced into the play, he is entering the Maxson home from the yard. He is well dressed and has the idea of himself as a musician, though he doesn't actually practice music. He knows that he will be successful, but he worries about possible judgment toward his lifestyle, particularly given his frequent need to borrow money from his father due to not having regular and stable work.
Lyon's greeting to his father, "Hey, Pop," is met with skepticism by Troy due to his awareness that his eldest son will ask him for money. From this exchange, Wilson establishes with the audience a sense that this, too, is a tense father-son relationship. Unlike his relationship with Cory—which is the result of his fear that his youngest son will surpass him in life by becoming the athlete that he wanted to become—Troy's relationship with Lyons is marked by Troy's belief in his own relative superiority. He has this sense of superiority because he is willing to live according to social dictates which say that a man must have a regular job.
Though Lyons's wife, Bonnie, works in the laundry room of a hospital, Lyons cannot bring himself to take on a job that will give him just enough to eat. He insists that he needs a reason to get out of bed in the morning, something that makes him want to live. Lyons differs from Troy in that he lives according to his desires, not according to any sense of obligation. He knows that others, particularly his father, judge him, but he refuses to accept society's refusal to allow a black man to live to his heart's content. Troy is a defeated man, while Lyons insists on maintaining some pride and hope of happiness. As a result, he refuses his father's offer to get him a job hauling trash, because he doesn't want to carry anyone's rubbish or punch a time clock.
Troy resents what he perceives as his son's aversion to regular hard work, as well as his willingness to take money from him. Lyons takes the money, and thinks that the money is due to him, because his father abandoned him and his mother when he was growing up. Thus, Lyons's relationship with Troy is also marked by a feeling that his father is feigning moral superiority and paternal authority, though he hasn't earned either right.