  Lately, the following problem has attracted a lot of attention in various
contexts: find the shortest factorisation $G=UU^-UU^-...U^{\pm}$ of a Chevalley
group $G=G(\Phi,R)$ in terms of the unipotent radical $U=U(\Phi,R)$ of the
standard Borel subgroup $B=B(\Phi,R)$ and the unipotent radical
$U^-=U^-(\Phi,R)$ of the opposite Borel subgroup $B^-=B^-(\Phi,R)$. So far, the
record over a finite field was established in a 2010 paper by Babai, Nikolov,
and Pyber, where they prove that a group of Lie type admits unitriangular
factorisation $G=UU^-UU^-U$ of length 5. Their proof invokes deep analytic and
combinatorial tools. In the present paper we notice that from the work of Bass
and Tavgen one immediately gets a much more general result, asserting that over
any ring of stable rank 1 one has unitriangular factorisation $G=UU^-UU^-$ of
length 4. Moreover, we give a detailed survey of triangular factorisations,
prove some related results, discuss prospects of generalisation to other
classes of rings, and state several unsolved problems. Another main result of
the present paper asserts that, in the assumption of the Generalised Riemann's
Hypothesis, Chevalley groups over the ring $\Int\Big[\displaystyle{1\over
p}\Big]$ admit unitriangular factorisation $G=UU^-UU^-UU^-$ of length 6.
Otherwise, the best length estimate for Hasse domains with infinite
multiplicative groups that follows from the work of Cooke and Weinberger, gives
9 factors.
