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.
BUTTER, YEAST, FLOUR, and SUGAR all fit the category "Baking staples". A bakery keeps these close. 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 / LAMP / NOTEBOOK / STAPLER
PENCIL, LAMP, NOTEBOOK, 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
Night sky
ORION / VENUS / NEBULA / METEOR
ORION, VENUS, NEBULA, and METEOR all fit the category "Night sky". Look up after sunset. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Musical directions
ADAGIO / PIANO / ALLEGRO / FORTE
ADAGIO, PIANO, ALLEGRO, and FORTE all fit the category "Musical directions". A score might tell a player these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.