Weinbaum factorizations of primitive words

Weinbaum [1] showed the following. Let $w$ be a primitive word and $a$ be letter in $w$. Then a conjugate of $w$ can be written as $uv$ such that $a$ is a prefix and $a$ suffix of $u$, but $v$ neither starts nor ends with $a$, and $u$ and $v$ have a unique position in $w$ as cyclic factors. The latter condition means that there is exactly one conjugate of $w$ having $u$ as a prefix and there is exactly one conjugate of $w$ having $v$ as a prefix. It is this condition which makes the result non-trivial.
We give a simplified proof for Weinbaum's result. Guided by this proof we exhibit quite different, but still simple, proofs for more general statements. For this purpose we introduce the notion of Weinbaum factor and Weinbaum factorization.