Given a K3 surface $X$, the cup product defines a non-degenerate even unimodular structure on the lattice $H^2(X,\mathbb{Z})$. Inside this lattice we have the Neron-Severi group $\text{NS}(X)$, which is also a primitive lattice. The rank of $\text{NS}(X)$, denoted by $\rho(X)$, is called the Picard number of $X$. The orthogonal complement of $\text{NS}(X)$ is by definition the transcendental lattice 
\begin{equation}
T(X):=\text{NS}(X)^\perp \subset H^2(X,\mathbb{Z}).
\end{equation}


In the note "Arithmetic of K3 surfaces" by Matthias Schutt, the author says that


"If $X$ is defined over some number field, the lattices of algebraic and transcendental cycles give rise to Galois representations of dimension $\rho(X)$ resp. $22-\rho(X)$."


Does he mean that the Galois representation arise from the etale cohomology $H^2\_{et}(X,\mathbb{Q}\_\ell)$ splits into the direct sum of two sub-representations with dimension $\rho(X)$ (associated to the algebraic cycles) and $22-\rho(X)$ (associated to the transcendental cycles)? 


I guess this statement might be true generally for algebraic surfaces. Could anyone explain it more carefully, and give a reference if possible?



Galois representations associated to the algebraic cycles and transcendental cycles of K3 surfaces
This is essentially the Tate conjecture for K3 surfaces which says that the image of the NS lattice tensored with $\mathbb{Q}\_l(1)$ in the etale cohomology $H^2\_{et}(X, \mathbb{Q}\_l(1))$ under the cycle class map is an invariant subspace under the action of the Galois group on the etale cohomology. 


