Let $(V,Q)$ be a quadratic module and $U$ is a subspace of $V$. Serre (A Course in Arithmetic, p. 29) claimed that the following is evident:
$U$ is isotropic $\iff$ $U\subset U^0$.
In other words, if $u.u=0$ for all $u\in U$, then $u.v=0$ for all $u,v\in U$. Why is this so?
Edit: Never mind, I got it. $u.v=0$ follows from $u.u=v.v=(u+v).(u+v)=0$. Please delete this question.