Prefixes of Coxeter elements are Coxeter elements of the reflection subgroups they generate

Let $(W,S)$ be a Coxeter system of finite rank, let $T$ be its set of reflections and let $c\in W$ be a Coxeter element. For $x$ in the interval $[1,c{]}_T$ of the absolute order, let $P(x)$ be the subgroup generated by the reflections $t{\le }_{T}x$. We prove that $x$ is a Coxeter element of $P(x)$ with respect to the canonical Coxeter generators of $P(x)$, that $P(x)$ is generated by the factors of any reduced reflection factorization of $x$, and that the Hurwitz action on the reduced reflection factorizations of $x$ is transitive. The proof reduces the general case to the case of a Coxeter system whose Coxeter matrix has entries in $\{1,2,3,\mathrm{\infty }\}$, which was treated by Hubery and Krause.