Pure spinors as a Koszul algebra

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


is Koszul.

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


It is therefore also Koszul.

We will denote:


In this case is the universal enveloping algebra:


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