The Dynkin index and sl2sl2-subalgebras of simple Lie algebras

Let gg be a simple Lie algebra. In this note, we provide simple formulae for the index of sl2sl2-subalgebras in the classical Lie algebras and a new formula for the index of the principal sl2sl2. We also compute the difference, D  , of the indices of principal and subregular sl2sl2-subalgebras. Our formula for D   involves some data related to the McKay correspondence for gg. Using the index of sl2sl2-subalgebras of classical Lie algebras, we obtain three series of interesting combinatorial identities parameterised by partitions.