4. As shown in Figure $1, A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is a cube. Any plane $\alpha$ is made perpendicular to the diagonal $A C^{\prime}$, such that $\alpha$ intersects each face of the cube. Let the area of the resulting cross-sectional polygon be $S$, and the perimeter be $l$. Then ( ).
(A) $S$ is a constant, $l$ is not a constant
(B) $S$ is not a constant, $l$ is a constant
(C) Both $S$ and $l$ are constants
(D) Neither $S$ nor $l$ are constants
4.B.

As shown in Figure 5, after cutting off two regular tetrahedra $A A^{\prime} B D$ and $C^{\prime}-D^{\prime} B^{\prime} C$ from a cube, a geometric body $V$ is obtained, with parallel planes $A^{\prime} B D$ and $D^{\prime} B^{\prime} C$ as its top and bottom bases. Each side face of $V$ is an isosceles right triangle, and each side of the polygonal section $W$ is parallel to a side of the base of $V$. When the side faces of $V$ are unfolded along the edge $A^{\prime} B^{\prime}$ and laid flat on a plane, a quadrilateral $A^{\prime} B^{\prime} B_{1} A_{1}$ is obtained, and the perimeter of the polygon $W$ unfolds into a line segment parallel to $A^{\prime} A_{1}$ (as shown in Figure 6 as $E^{\prime} E_{1}$). Clearly, $E^{\prime} E_{1} = A^{\prime} A_{1}$, so $l$ is a constant.

When $E^{\prime}$ is at the midpoint of $A^{\prime} B^{\prime}$, the polygon $W$ is a regular hexagon, and when point $E^{\prime}$ moves to point $A^{\prime}$, $W$ is a regular triangle. It is easy to know that the areas of a regular hexagon and a regular triangle with a constant perimeter $l$ are $\frac{\sqrt{3}}{24} l^{2}$ and $\frac{\sqrt{3}}{36} l^{2}$, respectively, so $S$ is not a constant.