Towards an explicit expression of the Seiberg-Witten map at all orders

The Seiberg-Witten map links non-commutative gauge theories to ordinary gauge theories, and allows to express the non-commutative variables in terms of the commutative ones. Its explicit form can be found order by order in the non-commutative parameter θ and the gauge potential A by the requirement that gauge orbits are mapped on gauge orbits. This of course leaves ambiguities, corresponding to gauge transformations, and there is an infinity of solutions. Is there one better, clearer than the others? In the abelian case, we were able to find a solution, linked by a gauge transformation to already known formulas, which has the property of admitting a recursive formulation, uncovering some pattern in the map. In the special case of a pure gauge, both abelian and non abelian, these expressions can be summed up, and the transformation is expressed using the parametrisation in terms of the gauge group.