Two abelian subalgebrasThe 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:

and are contained in and

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