i hope my question is not too trivial. Let's suppose we have a vector space $V$ with a unimodular quadratic form $q$ of signature $(m,n)$.

My question is: which is the maximum dimension of an istropic subspace of $V$?

I would say it is $min\{m,n\}$. This is why. Suppose $min\{m,n\}=m$, if $min\{m,n\}=n$ the proof is the same. Given $U$, a positive subspace of dimension $m$, i choose $u_1,\cdots, u_m$ an ortonormal basis of $U$. Then i consider $U^\perp$ which is negative and i choose $w_1,\cdots,w_n$ an ortonormal basis. So the vectors $\{u_1+w_1,u_2+w_2,\cdots,u_m+w_m\}$ span an isotropic subspace of dimension $m$.

But could there be an isotropic subspace of dimension more than $min\{m,n\}$?