I would start by trying to recreate the tongue and groove inserts for all three corners.
Cut out the triangle blocks all the same size. You will need 6.
Then build the frame those attach to. You will need 3 frames.
(Both sides will need a pilot hole going through for your Machine Screws & Wing nuts to fasten them together. I recommend getting them built, then putting them together and screwing the holes so you know they will line up.)
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*Use your nail gun for this process so you don’t have bulky screws everywhere. Add wood glue to help you keep things in place between every piece.
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Attach the frame piece with three screws to the outer board on the INSIDE of your shelf (board 2). Make sure your screws aren’t long enough to go through to the outside of your shelf.
On the other side, Just use your nail gun to attach the little triangle blocks to board 1, going through the triangle block at an angle, not from the outside of your shelf… You won’t want visible screws or nails anywhere on the outside of your shelf.
*Reinforce all pieces with wood glue so this lasts.