Lattices in Tate modules

Refining a theorem of Zarhin, we prove that given a $g$-dimensional abelian variety $X$ and an endomorphism $u$ of $X$, there exists a matrix $A\in {\mathrm{M}}_{2g}(\mathbb{Z})$ such that each Tate module $T_{\mathrm{\ell }}X$ has a ${\mathbb{Z}}_{\mathrm{\ell }}$-basis on which the action of $u$ is given by $A$, and similarly for the covariant Dieudonn'e module tensored with $\mathbb{Q}$ if over a perfect field of characteristic $p$.