Some remarks on the Maslov index

It is a classical fact that Wall's index of three Lagrangians in a symplectic space over a field k defines a $2$-cocycle μW on the associated symplectic group with values into the Witt group of k. Moreover, module the square of the fundamental ideal this is a trivial $2$-cocycle. In this work we revisit this fact from the viewpoint of the theory of Sturm sequences and Sylvester matrices developed by J.~Barge and J.~Lannes in their book ``Suites de Sturm, indice de Maslov et periodicite de Bott. (French) [Sturm sequences, Maslov index and Bott periodicity]'' Progress in Mathematics, 267. This allows us in particular to give an explicit formula for the coboundary associated to the mod I² reduction of the cocycle μW which is valid on any field of characteristic different from $2$