I'm reading Levin's note to learn the Tate conjecture for Abelian varieties over number fields. It seems that Tate's original paper and this note both use a weak finiteness hypothesis
There are infinitely many $B_n=A/G_n$ which are isomorphic, where $A\to B_n$ is an isogeny and all $B_n$ are equipped with polarizations.
I have understood that by considering the maximal isotropic subspace we can prove the key projective lemma
(Projection Lemma) Assume that $A$ has a polarization $\theta$.Let $W\subset V_\ell(A)$ be a $G_K$-stable subspace. Then, there exists a operator $u\in{\rm End}(A)\otimes Q_\ell$ such that $u^2=u$ and $uV_\ell(A)=W$.
But I don't know why this holds for Abelian varieties without polarizations. I found that in this note $\S$ 8.2.3 the author avoid the maximal isotropic subspace by proving that the finiteness hypothesis holds without assuming any polarized condition.
If I want to follow the first line, how do I reduce to the polarized cases?
I noticed that when we do the reduction
The map $${\rm Hom}(A, B)\otimes\mathbb{Q}_p\to{\rm Hom}(V_p(A),V_p(B))^{G_K}\quad(1)$$ is an isomorphism if and only if $${\rm End}(A)\otimes\mathbb{Q}_p\to{\rm End}(V_p(A))^{G_K}\quad(2)$$ is an isomorphism.
We actually prove a by-product that if $(2)$ holds for $A\times B$, then it holds for $A$ and $B$, right? Thus if $(2)$ holds for polarized Abelian varieties $C=A^4\times(A^\vee)^4$ (it seems that this holds if the projective lemma holds for $C$ and $C^2$), then we can deduce the Tate conjecture for general Abelian varieties $A$. This method seems to work but we do not get the general result for projective lemma.