Pure spinors as a Koszul algebra

It turns out that the algebra of pure spinors in 10D:

(19)

is Koszul.

The dual algebra is the algebra of covariant derivatives of the 10-dimensional supersymmetric Yang-Mills theory:

(20)

It is therefore also Koszul.

We will denote:

(21)

In this case is the universal enveloping algebra:

(22)

where is the Lie algebra formed by nested commutators of the letters .