Suppose you have a space $U$ of dimension $2n$, with symplectic forms $\Omega_1$ and $\Omega_2$. Is there a subspace $L$ which is Lagrangian for both forms? It´s not a problem I saw in any book, just curiosity.
It seems to me that the answer should be yes.
My try: If you find vectors $u,v$ such that $\Omega_1(u,v)\neq0\neq\Omega_2(v,u)$ and with Span$(u,v)^{\Omega_1}=$Span$(u,v)^{\Omega_2}$, then you can complete an argument by induction in the dimension. You can rephrase that in terms of the functions $\tilde{\Omega}_i:U\to U^*$ induced by $\Omega_i$, to get a problem about finding isotropic (for both $\Omega_1$ and $\Omega_2$) subspaces of dimension $2$ of $U$ which are invariant under $\tilde{\Omega}_2^{-1}\tilde{\Omega}_1$, but I wasn´t able to complete this argument.