Suppose $S$ is symplectic matrix with only real eigenvalues. I need to prove that $S$ has Lagrangian invariant subspace, i.e. there is $L$ - Lagrangian, such that $S(L) \subset L$.
I know that eigenvalues of symplectic matrix go in pair $\lambda, \frac{1}{\lambda}$. Also I know that two eigenvectors corresponding eigenvalues $\lambda, \mu$ such that $\lambda \mu \neq 1$ are orthogonal with respect to standart symplectic form in $\mathbb{R}^{2n}$. So I tried to prove that I can find $n$ such eigenvectors, then their span is required subspace, but I wasn't able to do it.
Any hints or references are welcome.
Thanks in advance.