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Arrange the lengths of all consecutive subwords of a word in a triangular grid. Under the folding condition, the boundary between length ascents and descents must contain a folding square. Its equal opposite vertices identify two letters that can be deleted. Induction proves the deletion condition for involutory generators.

The deletion condition for involutory generators says that whenever a wordis not reduced, there are indices $i<j$ such that deleting $s_i$ and $s_j$ leaves a word representing the same group element.

The exchange condition for a Coxeter group says that if $w=s_1\cdots s_m$ is reduced and $s\in S$ satisfies ${\ell }_S(sw)<{\ell }_S(w)$, thenfor some $i$. There is an equivalent right-handed form for ${\ell }_S(ws)<{\ell }_S(w)$.