= Solution
Let $\dim V=2n$ and let $V_+=\ker(S-I)$ and $V_-=\ker(S+I)$ be the two <eigenspaces>. The <involution> identity gives a <direct sum>, since
$$
v=\frac{v+Sv}{2}+\frac{v-Sv}{2},\qquad V=V_+\oplus V_-.
$$
For $u,v$ in either one of these <eigenspaces>, the <anti-symplectic involution> identity implies
$$
\Omega(u,v)=\Omega(Su,Sv)=-\Omega(u,v),
$$
so both are <isotropic subspaces of a symplectic vector space>. Such a subspace has dimension at most $n$: $W\subset W^\Omega$ and the <nondegenerate bilinear form> $\Omega$ gives $\dim W^\Omega=2n-\dim W$, where $W^\Omega$ is the <symplectic orthogonal complement>. As their dimensions add to $2n$, both have dimension $n$. Therefore
$$
\boxed{\operatorname{Fix}(S)=V_+\text{ is a Lagrangian subspace}.}
$$
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