Say we have two different digit transforms \(T_1\) and \(T_2\) that SWAP DIGITS, (later theorem non-simple tranforms and simple digit swaps), if we have a digit string \(D\) we know \[T_1(T_1(D))=D \\ T_2(T_2(D))=D\]
But does \[T_1(T_2(T_1(T_2(D))))=D \; ?\]
Drop the brackets and use commutators. Define a finite transform \[T_1abcde := ecbda \;\;|| \mathrm{s15,s23}\\ T_2abcde := dbcae \;\;|| \mathrm{s14,...}\\\]
Then \[T_1T_2abcde=T_1dbcae=ecbad \\ T_2T_1abcde=T_2ecbda=dcbea \\ T_2T_1T_2abcde=acbed \ne T_1abcde \\ T_1T_2T_1T_2abcde= dbcea \\ T_2T_1T_2T_1T_2abcde= ebcda \;\;|| \mathrm{s15} \\\\\\ T_3abcde := acbed \;\;|| \mathrm{s23,s45}\]
Thus have proven the trivial \(T_1T_2T_1T_1T_2T_1D=D\) kind of theme, any palindromic operator string is it’s own inverse.