We first draw a 4-fold rotation axes, then construct a circle centered with the axis and cross the three vertex on the end the four connected sides.

Second, we rotate the original octahedron and copy it.

Set one of the vertex on the end the four connected sides being moved around with the circle.

This is the rotated octahedron which will be projected on the original cube.

Select half of the rotated octahedron and the center face inside of it and then construct a convex hull composed with.

This constructed convex hull is a pyramid which is irregular.

The height of it is , where a is the side of the original octahedron.

This irregular pyramid is seemed as a splitter, which cut the original octahedron into component.

Because the interface between it and the original octahedron is the projection of the rotated octahedron, which was prooved by linear matrix equation.

We cut the original octahedron with the splitter which metioned in the Step 2.

The polygon with yellow faces is what we construct after splitting.

And the face will change when the vertex of rotated octahedron rotating, which shows the projected surface.

We see this resulted polygon as a component which combine a new octahedron because of symmetry.

Then we start to duplicate the component with symmetric properties of octahedron.

First, the polygon with red surfaces by the yellow one counter the plane centered the octahedron, which concludes 4 sides of the octahedron.

This is also the axisymmetric of reflection.

And then is the polygon with blue surfaces, which was duplicated with the same way but different plane.

Repeat the step above and then duplicate the green one.

These four component combine the top half of the octahedron.

Because of the resulted polygon mentioned above, we can easily duplicate the buttom half of the octahedron by point symmetric.

First if the blue one, which is duplicated by the other blue one marked with stripes.

And then is the yellow one, using the same method mentioned above.

Also the polygon with green surfaces.

Finally is the polygon with red surfaces.

Then we can construct the final result: the projection of a rotated octahedron on the original octahedron.