The global symmetry algebra is . Let us for now neglect the reality structure, and just study the complex Lie superalgebra .
We will start by making some observations about . This can be realized as supermatices acting in . Let us introduce additional structure: a pair of nondegenerate antisymmetric forms:
To abuse notations, we will denote them with the same letter . Such a form establishes an isomorphism of associative algebras:
In other words:
Then, the following map:
is a -automorphism of .
Elements of grading 3 and 1 are:
Properties of trace:
Without loss of generality we can assume:
Elements of grade 2 can be parametrized explicitly:
These matrices have an interesting property:
and in particular: