4. As shown in the figure, in the right dihedral angle $D-A B-E$, quadrilateral $A B C D$ is a square with a side length of 2, $A E=B E$, and $F$ is a point on $C E$ such that $B F \perp$ plane $A C E$. The size of the dihedral angle $B-A C-E$ is $\qquad$
4. $\arccos \frac{\sqrt{3}}{3}$ Detailed Explanation: Establish the system $A E \perp B F, A E \perp B C$ to get $A E \perp B E$, which means $\triangle A E B$ is an isosceles right triangle, so point $E(1,0,0)$, the normal vector of plane $B A C$ is $\boldsymbol{n}_{1}=(1,0,0)$, the normal vector of plane $A C E$ is $\boldsymbol{n}_{2}=(1,-1,1)$, thus $\arccos \frac{\sqrt{3}}{3}$ equals the required angle.