Kuga–Satake abelian varieties and $l$-adic representations

Let $J$ be a Kuga–Satake abelian variety defined over a number field $k\hookrightarrow\mathbf C$. Assuming a certain arithmetic condition on the canonical field $K$ associated to $J\otimes_k\mathbf C$, we prove the Mumford–Tate conjecture concerning the Lie algebra of the image of the $l$-adic representation in the one-dimensional cohomology of $J$.