Subspaces in associated to a pair of pure spinors

 (9)
Suppose that we are also given a pair of pure spinors and . Then we can refine the decomposition of Eq. (9) by further splitting each , as a linear space, into a direct sum of linear subspaces.

#### 1Notations

For any , we will denote:
where was defined in Eq. (4).

#### 2Tangent and normal space to pure spinor cones

The projector
 (10)
 (11)
where is adjusted to satisfy (11). In fact is the projection to the tangent space along the space which is orthogonal to with respect to the metric defined by :
The projector is defined similarly. We have the following exact sequence of linear spaces:

#### 3Refinement of

Consider the decomposition:
Here is a 4-dimensional subspace -orthogonal to and commuting with :
and is -orthogonal to and commuting with :

#### 4Refinement of and

Similarly we can split and as follows:
Therefore, as a linear space:

#### 5More definitions

For and , let denote the map:

This is a direct sum of two completely independent linear maps: a map and a map

For a pair we decompose:
where a special representative of the cokernel is used:
Similarly, any (assumed to be both TL and STL) can be decomposed:
Explicitly: