Symmetries
The global symmetry algebra is . Let us for now neglect
the reality structure, and just study the complex Lie superalgebra .
1 Some facts about
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:
We observe:
Properties of trace:
2 Case
Without loss of generality we can assume:
Elements of grade 2 can be parametrized explicitly:
 

These matrices have an interesting property:
Therefore:
and in particular: