The ordinary Coxeter presentation is the empty-J case

Given: A Coxeter system as in the Statement, with $J=\mathrm{\varnothing }$ and distinct vertices $a,b$ when a pair is considered. The length $\mathrm{\ell }$ is the minimum number of simple generators in a word representing an element. A standard expression is a product of elementary arrows whose Coxeter lengths sum to the length of its label. Coxeter-system, roots, length and weak-order conventions Atomic braid relations

[F1] The empty set is its only associate and $v_{a,\mathrm{\varnothing }}=s_a$ for every vertex $a$. Empty-subset wall crossings are simple reflections

[F2] A Brink–Howlett arrow $I\to K$ is labelled by an element carrying the simple-root set of $K$ onto that of $I$; composition multiplies labels from left to right. Brink–Howlett groupoid

[F3] For $T\subseteq {\mathrm{\Gamma }}_0$, $W_T$ is generated by the $s_t$ with $t\in T$, and ${\mathrm{\Phi }}_T=\{w{\alpha }_t:w\in W_T,t\in T\}$. In particular ${\mathrm{\Phi }}_{\mathrm{\varnothing }}=\mathrm{\varnothing }$. Standard parabolic subgroups and parabolic roots

[F4] For distinct $a,b$, the two-generator construction is defined when ${\mathrm{\Phi }}_{\{a,b\}}$ is finite. In that case its label is $q=w_D{w}_{D\cap \mathrm{\varnothing }}$, where $D$ is the union of components of the diagram on $\{a,b\}$ meeting $\{a,b\}$; $D$ is spherical, meaning $W_D$ is finite. It has exactly two standard expressions, one beginning with each elementary arrow. Finite relative longest element without spherical ambient rank Two-generator minimal common left multiple

[F5] The rank-two parabolic is the Coxeter group for the restricted matrix, with intrinsic length equal to ambient length. For finite $m(s_a,s_b)=m$, it has $2m$ elements and its unique longest element has length $m$ and is the alternating word of length $m$ beginning with $s_a$; the same assertion holds with $a,b$ exchanged. For infinite $m(s_a,s_b)$ the alternating words of increasing length are distinct. The order of $s_a{s}_b$ is exactly $m(s_a,s_b)$, with infinity interpreted as infinite order. Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank The rank-two half-space alternative and the chamber-length induction (Pₙ), (Qₙ) The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness

[F6] The Brink–Howlett presentation has the elementary arrows as generators and the atomic braid relations and elementary inverse-pair relations as defining relations. An atomic relation is the equality of the two standard expressions supplied for a distinct pair with finite relative root difference. Brink–Howlett presentation by atomic and quadratic relations Atomic braid relations

Proof technique: direct.

Goal: identify the arrows and all defining relations when $J=\mathrm{\varnothing }$.