  We propose a novel construction for the well-known prefer-max De Bruijn
sequence, based on the cycle joining technique. We further show that the
construction implies known results from the literature in a straightforward
manner. First, it implies the correctness of the onion theorem, stating that,
effectively, the reverse of prefer-max is in fact an infinite De Bruijn
sequence. Second, it implies the correctness of recently discovered shift rules
for prefer-max, prefer-min, and their reversals. Lastly, it forms an
alternative proof for the seminal FKM-theorem.
