Let $I$ be the Dynkin diagram vertex set and $K$ be a proper nonempty subset of it.

I saw in the paper "Preprojective algebras and partial flag varieties" by Geiss, Leclerc and Schroer the following statement:

"We can write $w_0 = w^K_0 v_K$ with $\ell(w^K_0) + \ell(v_K) = ℓ(w_0).$ Therefore there exist reduced words $i$ for $w_0$ starting with a factor $(i_1,...,i_{r_K})$ which is a reduced word for $w^K_0$."

I could run this through examples, like writing the longest element $w_0=s_3s_2s_3s_1s_2s_3s_1s_2s_1$ of type $B$ as $w_0=\boldsymbol{s_1s_2s_1} {s_3s_2s_1s_3s_2s_3}$, if $K=\{3\}$.

I was wondering if there is an algorithm showing how to do this in general, at least if the size of $K$ is known, like $|K|=1$.