This is the most simple form of the following stronger statement:

• Two words commute if and only if they satisfy a nontrivial relation. This, in turn, is the GraphLemma for two words.

Yet more general is the following fact which is connected to the PeriodicityLemma:

• Let w_u and w_v be two words generated by u and v, where w_u starts with u and w_v starts with v. If w_u and w_v have a common prefix of length at least |u|+|v|-GCD(|u|,|v|), then u and v commute.

-- StepanHolub - 10 Sep 2011