On unique factorizations of primitive words

We give a short proof of a result by Weinbaum [Unique sunwords in nonperiodic words, Proc. Amer. Math. Soc. 109(3) (1990) 615–619] stating that each primitive word of length at least 2 has a conjugate w′=uv such that both u and v have a unique position in the cyclic word of w. We emphasize the connection of Weinbaum's result to the Critical Factorization Theorem.