Simultaneous bosonic and fermionic T-dualization of the type II superstring theory -- Buscher approach and double space representation

In this article I consider type II superstring in the pure spinor formulation with constant background fields in the context of T-dualization. First I prove that bosonic and fermionic T-dualization commute using already known T-dual transformation laws for bosonic and fermionic T-dualization. Consequently, the T-dual transformation laws of the full T-dualization are obtained. At the end the full T-dualization is realized in double space and it is showed that Buscher procedure and double space approach are equivalent in this specific case.