The following theorem was proved independently by Deodhar and by Dyer:

Let $(W, S)$ be a Coxeter group. Let $V$ be a subgroup of $W$ generated by reflections. Then there is a set $R$ of reflections in $V$ so that $(V, R)$ is a Coxeter group. $R$ can be characterized uniquely as follows: Let $T$ be the set of reflections in $W$ and, for $w \in W$, let $in(w)$ be the set of inversions of $w$. An element $t \in T$ lies in $R$ if and only if $in(t) \cap V = \{ t \}$.

Even when $S$ is finite, $R$ can be infinite! If I haven't messed up, one example is to take $W$ to be the Coxeter group on generators $p$, $q$, $r$, with no relations other than $p^2=q^2=r^2=e$. Let $V$ be the subgroup generated by $$\cdots qpqprpqpq, qpqrqpq, qprpq, qrq, r, prp, pqrqp, pqprpqp, pqpqrqpqp, \cdots.$$ If I haven't messed up, $V$ is an infinite rank subgroup of $W$, with the set $R$ being the list of generators above.

Since these two papers have 86 citations in Mathscinet between them, I'd say this result is useful.

The following theorem was proved independently by Deodhar and by Dyer:

Let $(W, S)$ be a Coxeter group. Let $V$ be a subgroup of $W$ generated by reflections. Then there is a set $R$ of reflections in $V$ so that $(V, R)$ is a Coxeter group. $R$ can be characterized uniquely as follows: Let $T$ be the set of reflections in $W$ and, for $w \in W$, let $in(w)$ be the set of inversions of $w$. An element $t \in T$ lies in $R$ if and only if $in(t) \cap V = \{ t \}$.

Even when $S$ is finite, $R$ can be infinite! If I haven't messed up, one example is to take $W$ to be the Coxeter group on generators $p$, $q$, $r$, with no relations other than $p^2=q^2=r^2=e$. Let $V$ be the subgroup generated by $$\cdots, qpqprpqpq, qpqrqpq, qprpq, qrq, r, prp, pqrqp, pqprpqp, pqpqrqpqp, \cdots.$$ If I haven't messed up, $V$ is an infinite rank subgroup of $W$, with the set $R$ being the list of generators above.

Since these two papers have 86 citations in Mathscinet between them, I'd say this result is useful.

The following theorem was proved independently by Deodhar and by Dyer:

Let $(W, S)$ be a Coxeter group. Let $V$ be a subgroup of $W$ generated by reflections. Then there is a set $R$ of reflections in $V$ so that $(V, R)$ is a Coxeter group. $R$ can be characterized uniquely as possible: Let $T$ be the set of reflections in $W$ and, for $w \in W$, let $in(w)$ be the set of inversions of $w$. An element $t \in T$ lies in $R$ if and only if $in(t) \cap V = \{ t \}$.

Even when $S$ is finite, $R$ can be infinite! If I haven't messed up, one example is to take $W$ to be the Coxeter group on generators $p$, $q$, $r$, with no relations other than $p^2=q^2=r^2=e$. Let $V$ be the subgroup generated by $$\cdots qpqprpqpq, qpqrqpq, qprpq, qrq, r, prp, pqrqp, pqprpqp, pqpqrqpqp, \cdots.$$ If I haven't messed up, $V$ is an infinite rank subgroup of $W$, with the set $R$ being the list of generators above.

Since these two papers have 86 citations in Mathscinet between them, I'd say this result is useful.

The following theorem was proved independently by Deodhar and by Dyer:

Let $(W, S)$ be a Coxeter group. Let $V$ be a subgroup of $W$ generated by reflections. Then there is a set $R$ of reflections in $V$ so that $(V, R)$ is a Coxeter group. $R$ can be characterized uniquely as follows: Let $T$ be the set of reflections in $W$ and, for $w \in W$, let $in(w)$ be the set of inversions of $w$. An element $t \in T$ lies in $R$ if and only if $in(t) \cap V = \{ t \}$.

Even when $S$ is finite, $R$ can be infinite! If I haven't messed up, one example is to take $W$ to be the Coxeter group on generators $p$, $q$, $r$, with no relations other than $p^2=q^2=r^2=e$. Let $V$ be the subgroup generated by $$\cdots qpqprpqpq, qpqrqpq, qprpq, qrq, r, prp, pqrqp, pqprpqp, pqpqrqpqp, \cdots.$$ If I haven't messed up, $V$ is an infinite rank subgroup of $W$, with the set $R$ being the list of generators above.

Since these two papers have 86 citations in Mathscinet between them, I'd say this result is useful.