In the proof of the signature property of Maslov index, one uses the fact that if $S$ is a symmetric matrix whose norm is less than $2\pi$, then the matrix $\exp(J_0S)$ does not admit $1$ as eigenvalue (in ``Morse theory and Floer homology'' by Michele Audin p.198, and here p.39), where $J_0$ is the standard complex structure $$J_0=\begin{pmatrix}0&-Id\\Id&0 \end{pmatrix}$$
Could you expain why is that?