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.
SEED, ATOM, BYTE, and PIXEL all fit the category "Tiny units". Smallness connects them. 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 / MAT / KETTLEBELL
ROPE, BENCH, MAT, and KETTLEBELL 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 with lanes
TRACK / POOL / BOWLING / ROAD
TRACK, POOL, BOWLING, and ROAD all fit the category "Things with lanes". Movement is organized into lanes. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Book parts
CHAPTER / COVER / INDEX / SPINE
CHAPTER, COVER, INDEX, and SPINE all fit the category "Book parts". They help make up a printed book. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.