I am interested in the Christoffel symbol. So I found Christoffel's 1869 paper and The Road to Reality by Roger Penrose. These days, the Christoffel symbol has been established as a covariant derivative, but Christoffel did not know the concept of covariant derivative and parallel transport.
So, I wondered what concept Christoffel used to create the Christoffel symbol at the time.
When I read the book The Road to Reality, I think that Lie algebra was use. What exactly is the concept behind the Christoffel symbol?