Minimal coset representatives of $S_2$ in $S_3$

The parabolic subgroup. $W_J=⟨J⟩=⟨s_1⟩=\{1,s_1\}$: the definition of $W_J$ as the subgroup generated by $J$ is [F5] and its identity with $⟨s_1⟩$ is immediate for $J=\{s_1\}$ [F2]; the relator $s_1^2$ of the presentation gives $s_1^2=1$ [F5], so $⟨s_1⟩\subseteq \{1,s_1\}$, and $s_1\mathrm{\ne }1$ because $\mathrm{\ell }(s_1)=1$ [F6], so $⟨s_1⟩$ has the two distinct elements $1,s_1$; it is a group of order $2$, hence isomorphic to $S_2$.

The left cosets. The left cosets $a{W}_J$ are $\{1,s_1\}$, $s_2{W}_J=\{s_2,s_2{s}_1\}$ and $s_1{s}_2{W}_J=\{s_1{s}_2,s_1{s}_2{s}_1\}$, disjoint and exhausting $W$ by the six distinct elements listed in [F3]; their minimal elements are $1$ (length $0$), $s_2$ (length $1$, versus $\mathrm{\ell }(s_2{s}_1)=2$) and $s_1{s}_2$ (length $2$, versus $\mathrm{\ell }(s_1{s}_2{s}_1)=3$), unique by the left-coset half of [F1]. For these three elements $d$ the right-handed additivity of [F1] gives $\mathrm{\ell }(d{s}_1)=\mathrm{\ell }(d)+\mathrm{\ell }(s_1)=\mathrm{\ell }(d)+1$, namely $\mathrm{\ell }(s_1)=1$, $\mathrm{\ell }(s_2{s}_1)=2=1+1$ and $\mathrm{\ell }(s_1{s}_2{s}_1)=3=2+1$, as asserted; equivalently the left coset representatives satisfy the characterisation $\mathrm{\ell }(d{s}_1)>\mathrm{\ell }(d)$ of [F1].

The right cosets and their minimal elements. For $W_J=\{1,s_1\}$ the sets $W_{J}a=\{ua:u\in W_J\}$ of [F4], with $a\in W$, can be enumerated using the six-element list of [F3] they are $W_J\cdot 1=\{1,s_1\}$, $W_J{s}_2=\{s_2,s_1{s}_2\}$ and $W_J{s}_2{s}_1=\{s_2{s}_1,s_1{s}_2{s}_1\}$, which are disjoint and exhaust $W$. Reading the lengths off [F3], the minimum of $\mathrm{\ell }$ is $0$ on $\{1,s_1\}$ (attained at $1$), $1$ on $\{s_2,s_1{s}_2\}$ (attained at $s_2$) and $2$ on $\{s_2{s}_1,s_1{s}_2{s}_1\}$ (attained at $s_2{s}_1$); by [F1] each coset has a unique element of minimal length, so the minimal representatives are $1$, $s_2$ and $s_2{s}_1$ with lengths $0$, $1$ and $2$.

The descent characterisation and length additivity. For the three minimal representatives $d=1$, $d=s_2$ and $d=s_2{s}_1$, the table of [F3] gives $\mathrm{\ell }(s_1\cdot 1)=\mathrm{\ell }(s_1)=1>0=\mathrm{\ell }(1)$, $\mathrm{\ell }(s_1{s}_2)=2>1=\mathrm{\ell }(s_2)$ and $\mathrm{\ell }(s_1{s}_2{s}_1)=3>2=\mathrm{\ell }(s_2{s}_1)$, so each of them satisfies the characterisation $\mathrm{\ell }(s_{1}d)>\mathrm{\ell }(d)$ of [F1]. The additivity identity of [F1] for $u=s_1\in W_J$ reads $\mathrm{\ell }(s_{1}d)=\mathrm{\ell }(s_1)+\mathrm{\ell }(d)$ for the minimal $d$; at $d=s_2$ this is $\mathrm{\ell }(s_1{s}_2)=1+1=2$ and at $d=s_2{s}_1$ it is $\mathrm{\ell }(s_1{s}_2{s}_1)=1+2=3$, the two values asserted in the statement.

Collected. The example enumerates the three right cosets and the three left cosets of $W_J=\{1,s_1\}$ in $W\cong S_3$, identifies their unique minimal representatives with the lengths predicted by the parabolic theorem, verifies the descent characterisations on both sides and the length-additivity identities $\mathrm{\ell }(ud)=\mathrm{\ell }(u)+\mathrm{\ell }(d)$ and $\mathrm{\ell }(du)=\mathrm{\ell }(d)+\mathrm{\ell }(u)$ for $u=s_1$ (steps 2.1, 3.1, 1.2).