Under certain, special rules, a king and rook can move simultaneously in a castling move.
The following conditions must be met:
When castling, the king moves two squares towards the rook, and the rook moves over the king to the next square, i.e., white's king on e1 and rook on a1 move to: king c1, rook d1 (long castling), white's king on e1 and rook on h1 move to: king g1, rook f1 (short castling), and similar for black.
position before and after castling: white short, and black long
Neither white nor black may castle: white is in check, and the black king may not move over d8
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