4. Given a right triangle $\triangle A B C$ with the altitude from the right angle to the hypotenuse being $C D$, and $A D=\frac{1}{3} A B$. If $\triangle A C D$ is rotated around $C D$ to $\triangle A_{1} C D$, such that the dihedral angle $A_{1}-C D-B$ is $60^{\circ}$. Then the angle between the skew lines $A_{1} C$ and $A B$ is $\qquad$ (express the answer using inverse trigonometric functions).
4. $\arccos \frac{\sqrt{3}}{6}$