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.
CHILI, PEPPER, WASABI, and GINGER all fit the category "Spicy things". They can wake up your taste buds. 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 ring
BELL / PHONE / ALARM / CASHIER
BELL, PHONE, ALARM, and CASHIER all fit the category "Things that ring". Each can make a ringing sound. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things in a gym
ROPE / BENCH / KETTLEBELL / MAT
ROPE, BENCH, KETTLEBELL, and MAT all fit the category "Things in a gym". Training rooms are their habitat. 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
ALGAE / FIREWORK / ROSE / TULIP
ALGAE, FIREWORK, ROSE, and TULIP 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.