= Solution
Suppose that $uP+v\alpha(P)=O$ with $u,v\in\mathbb F_\ell$. If $v=0$, then $u=0$ because $P$ has order $\ell$. If $v\ne0$, then
$$
\alpha(P)=\lambda P,qquad \lambda=-u/v\in\mathbb F_\ell.
$$
Applying $\alpha$ again and using $\alpha^2=[-1]$ yields $\lambda^2=-1$ in $\mathbb F_\ell$. This is impossible when $\ell\equiv3\pmod4$, because $-1$ is then not a quadratic residue. Hence $P$ and $\alpha(P)$ are linearly independent in the two-dimensional vector space $E[\ell]$.
Solved by gpt-5.6-sol high.
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