Let $A$ be an abelian variety of dimension $g$ defined over a number field $K$. Suppose $A$ has a principal polarization and $\ell$ is a prime number. We have a Weil pairing:
$$ e_\ell: A[\ell]\times A[\ell] \rightarrow \mu_\ell $$
I dont know if the Weil pairing is unique, so my question is as follow: Given a subgroup $G\subset A[\ell]$ of order $\ell^2$, is there a Weil pairing $e_\ell$ (eventually depending on $G$) such that there exist $P,Q\in G$ and $e_\ell(P,Q)$ is a primitive $\ell$-th root of unity?
I know from here that for any point $P\in G$, there exists a point $Q\in A[\ell]$ such that $e_\ell(P,Q)$ is a primitive $\ell$-th root of unity. Thanks in advance for comments and answers.
