4. A chord $A B$ is drawn through the right focus $F_{2}$ of the ellipse $\frac{x^{2}}{2}+y^{2}=1$ with an inclination angle of $45^{\circ}$. Then $|A B|$ is ( ).
(A) $\frac{2 \sqrt{6}}{3}$
(B) $\frac{4 \sqrt{6}}{3}$
(C) $\frac{4 \sqrt{2}}{3}$
(D) $\frac{4 \sqrt{3}}{3}$
4. C.

It is known that the right focus of the ellipse is $(1,0)$. Then $l_{A B}: y=x-1$.
Substituting the above equation into the ellipse equation, we get
$$
\begin{array}{l}
3 x^{2}-4 x=0 \Rightarrow x_{1}=0, x_{2}=\frac{4}{3} \\
\Rightarrow|A B|=\sqrt{2\left(x_{1}-x_{2}\right)^{2}}=\frac{4 \sqrt{2}}{3} .
\end{array}
$$