Equivalence of finite word metrics

For two finite symmetric choices of generating set of a group $S,T$, let $K=\max (\{1\}\cup \{|s{|}_T:s\in S\})$ and $L=\max (\{1\}\cup \{|t{|}_S:t\in T\})$. Replacing letters by their fixed words gives $|g{|}_T\le K|g{|}_S$ and $|g{|}_S\le L|g{|}_T$. Thus the identity is a bilipschitz equivalence between the word metrics. Finiteness is essential for these maxima; an infinite generating set can collapse whole subgroups to diameter one.