On the presentation of the Grothendieck-Witt group of symmetric bilinear forms over local rings

We prove a Chain Lemma for inner product spaces over commutative local rings R with residue field other than F2 and use this to show that the usual presentation of the Grothendieck-Witt group of symmetric bilinear forms over R as the zero-th Milnor-Witt K-group holds provided the residue field of R is not F2.