The question comes from Lemma 13. It is stated as follows.
Let $A$ be an associative ring with $1$, and $n\geq2$ an integer. Assume that either $n\geq3$ or $n=2$, and $1$ is the sum of two units in $A$. If $\beta=\lambda\rho\in\mathrm{GL}_n(A)$ is a product of a lower triangular matrix $\lambda$ and an upper triangular matrix $\rho$ in $\mathrm{GL}_nA$, then $\beta$ is a product of two commutator.
I am particularly impressed with this result. However, I feel it doesn't seem right. For example, if $A$ is a field and the determinant of $\beta$ is not 1 then this cannot happen. I also read the proof and got the idea of the proof. But I think it will be true for a product of the upper and lower triangular matrices that have a coefficient on the main diagonal of $1$. I think of this because I am not sure of the similarity of the matrices in the first three lines of the proof. Furthermore, the authors make a distinction between triangular matrices and triangular matrices that have a coefficient on the diagonal of $1$. Please see help me.