In this answer, it is claimed that the determinant of a Cartan matrix is same as its "index" of the root lattice in the weight lattice.
Firstly, what is the equation which describes the definition of this index? I searched for it in Google and books but couldn't find it. Moreover I got to learn about a different index in this wiki page which does not match the determinant as clearly shown in the table. The term "index" is arguably overused in literature, so it is perhaps not a good idea to naively try to equate the word "index" on two different wiki pages: but still I am comparing these because both of them appear beside determinant of Cartan matrix. There is even a Dynkin index. What exactly is meant by the word "index" in the context where it matches the determinant of the Cartan matrix?
Secondly, in this post Torsten Schoeneberg had said that the calculations are essentially the same as in here, but I don't see the equivalence (partly because I don't know the definition of "index"). So I want to know how to prove that the determinant of the Cartan matrix matches the "index"?
Maybe this is trivial but I am a beginner, so I am requesting some assistance in this. Thanks in advance.