  We give an algebraic proof of the independence of Coxeter moves involved in
the construction of positive representations of split-real quantum groups, thus
completing a gap in the original construction. To do this, we propose a new
quantized version of Lusztig's Injectivity Lemma in the language of quantum
cluster algebra, the proof of which by Tits' Lemma reduces to calculations
involving sequences of Coxeter moves forming rank 3 cycles. We give a new,
constructive proof of Tits' Lemma, and provide the required explicit
computation of the quantum cluster mutations under these rank 3 cycles using
certain cluster algebraic tricks via universally Laurent polynomials.
