To attach the seat of the chair to the frame, I chiselled out two perpendicular grooves on the bottom of the seat. This is the kind of work that a CNC router would do very well. However, once I found the right technique, it went pretty quickly. I just had to make sure to be very accurate with the tracing at the beginning because I am counting on the snug fit between the grooves and the frame to hold the seat in place.
First, using the flat side of the chisel, I marked the sides of the groove down to the depth I wanted.
Then, using the bevelled side of the chisel, I knocked out the material from the middle.