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 \(\mathbb {F}_2\) ${\mathbb{F}}_2$ and use this to show that the usual presentation of the Grothendieck–Witt group of symmetric bilinear forms over R as the zeroth Milnor–Witt K-group holds provided the residue field of R is not \(\mathbb {F}_2\) ${\mathbb{F}}_2$ .