Father Brown refers to his battered old bicycle as Bucephalus. That was the name of Alexander the Great's horse.
This episode takes its title directly from the short story, "The Three Tools of Death," from The Innocence of Father Brown. The characters are roughly the same in name and personality, although more backstory is provided in the episode. The public nature of the incident is reduced a bit for television, but otherwise this is a fairly faithful retelling of the original story.
The piano work performed by Alice's mother is Gymnopedie No. 3 by Erik Satie.