Sand the 10 inch-by-10 inch board (the rectangular seat) to a smooth sheen, first with the 80 grit to remove burrs and splinters followed by the 220 grit. Blow away any sand dust. Set aside.
Sand each of the four 4-by-4 inch pieces if necessary. Otherwise, turn the seat section over to face the underside. Turn a 4-by-4 inch piece on its side and place the the end vertically into one corner of the seat; trace around the base of the piece to indicate proper leg placement. Repeat for the three remaining corners to map out the location for all four stool legs.
Position one of the legs into place on the seat's underside and drill through the top of the seat, into the top of the leg, to secure the leg in place. Use two to three screws to create a secure joint. Repeat for the remaining three legs. Place a small bead of wood glue around the leg/seat joint to create additional stability.
Screw each 8 inch piece around the inside perimeter of the legs, screwing from outside of the legs to the inside wood slats. This helps add even more strength and stability. Apply a small bit of wood putty over the exposed screw heads to hide them. Paint, stain or clear coat the entirety of the stool prior to using.