Place the chair on a work surface so that the split part is facing you. Slide a putty knife into the split.
Pry gently on the split to open it just enough so that you can saturate the inside of the split with wood glue. Squeeze wood glue into the split and remove the putty knife.
Position a wood clamp over the split part. Tighten the clamp and wipe off any excess glue that seeps out from the split with a damp cloth. Allow the glue to dry for 24 hours and remove the clamp.
Sand the repaired split with a fine-grit sanding block until the repair is blended with the surface of the wood.
Apply a light coat of matching wood finish over the repaired area with a natural-bristle brush. Allow the finish to dry for four hours and apply a second coat if necessary.