  We show that the Andr\'{e} motive of a hyper-K\"{a}hler variety $X$ over a
field $K \subset \mathbb{C}$ with $b_2(X)>6$ is governed by its component in
degree $2$. More precisely, we prove that if $X_1$ and $X_2$ are deformation
equivalent hyper-K\"{a}hler varieties with $b_2(X_i)>6$ and if there exists a
Hodge isometry $f\colon H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})$, then the
Andr\'e motives of $X_1$ and $X_2$ are isomorphic after a finite extension of
$K$, up to an additional technical assumption in presence of non-trivial odd
cohomology. As a consequence, the Galois representations on the \'{e}tale
cohomology of $X_1$ and $X_2$ are isomorphic as well. We prove a similar result
for varieties over a finite field which can be lifted to hyper-K\"{a}hler
varieties for which the Mumford--Tate conjecture is true.
