Two abelian subalgebras

The even subalgebra of is a direct sum of two Lie algebras:


(The real form would be and .) Schematically, in the -block notations:

The in the upper right corner and in the lower left corner are both odd abelian subalgebras isomorphic to .

The grading which was used to construct the -model action is not very transparent in these notations. Notice that:

  1. and are contained in and

  2. is some linear combination of and and is the linear combination with the opposite relative sign between and