Given bilinear forms $B_1 , B_2 : V \times V \to k$, where $V$ is a finite dimensional vector space over the field $k$, we say that $W \subset V$ is an isotropic subspace of $V$ with respect to $B_1$ if $B_1|_W =0$ (restriction of $B_1$ to $W$ is zero).
How does one find a maximal common isotropic subspace of $B_1$ and $B_2$? I tried computing maximal isotropic subspaces of $B_1$ and $B_2$ and taking their intersection, it is a common isotropic subspace, but not maximal. (And often it is just $\{0\}$ )
Feel free to give partial results (e.g. specialize $k$). Thanks a lot!