Do involutions always stabilize some transverse lagrangians?

Question

Let $V$ be a vector space of dimension $2n\geq 4$ over a field $F$ of characteristic distinct from $2$. Assume that $V$ is equipped with a nondegenerate alternating form $b$. Let $Sp(V)$ denote the symplectic group with respect to $b$. A totally isotropic subspace $L$ of dimension $n$ is called lagrangian. An element $\sigma \in Sp(V)$ such that $\sigma^2=1$ is called symplectic involution. Is it true that given two symplectic involutions $\sigma_1, \sigma_2$ there are two lagrangians $L_1,L_2$ such that $V=L_1\oplus L_2$ and $\sigma_1(L_i)=L_i$, $\sigma_2(L_i)=L_i$ for $i=1,2$?