Relations in In and InWq in characteristic 2

Let K be a field of characteristic 2. We give natural presentations of the subgroups In(K) of the Witt ring W(K) of K and the subgroups InWq(K) of the Witt group Wq(K) of K. Our results generalize the results of Arason and Elman in [J.K. Arason, R. Elman, Powers of the fundamental ideal in the Witt ring, J. Algebra 239 (2001) 150–160], where the characteristic is assumed to be different from 2.