The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
FALCON, ARROW, COMET, and SPRINTER all fit the category "Fast things". Speed is the common thread. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Desk items
PENCIL / NOTEBOOK / LAMP / STAPLER
PENCIL, NOTEBOOK, LAMP, and STAPLER all fit the category "Desk items". They belong within arm's reach at work. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things that bloom
TULIP / ROSE / FIREWORK / ALGAE
TULIP, ROSE, FIREWORK, and ALGAE all fit the category "Things that bloom". Each can bloom in its own way. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Stage terms
SCENE / PROP / ENCORE / CURTAIN
SCENE, PROP, ENCORE, and CURTAIN all fit the category "Stage terms". A theater audience might notice them. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.