Cut the mitered ends of the box on a table saw. Raise the blade up above the table and tilt it to a 45-degree angle. Use a bevel gauge to make sure that the angle is exactly 45 degrees.
Square up the cut and fasten it to an auxiliary fence. Use a scrap fence if the cuts are too small. Miter cut each end of the parts and clamp them.
Locate the splines at or near the inside of the corner of the joint so you can ensure the maximum amount of strength needed. Using a stop block, mark one miter cut where the spline will go. Clamp the block to the miter gauge fence.
Adjust the height of the blade to cut half the spline width into each end that is mitered. Set the blade so the cut will be a little less than 3/8 inch deep. Make a test cut before actually cutting the wood.
Tear the strips of the spline stock off of the edge on one of the long pieces of scrap wood. Remember to match the width of the spline slots so there is room between the fence and blade.
Test the strip to make sure if fits correctly. Glue in place.