  We perform a systematic bootstrap analysis of four-point one-loop Mellin
amplitudes for super gluons in $\mathrm{AdS}_5\times\mathrm{S}^3$ with
arbitrary Kaluza-Klein weights. The analysis produces the general expressions
for these amplitudes at extremalities two and three, as well as analytic
results for many other special cases. From these results we observe remarkable
simplicity. We find that the Mellin amplitudes always contain only simultaneous
poles in two Mellin-Mandelstam variables, extending a previous observation in
the simplest case with the lowest Kaluza-Klein weights. Moreover, we discover a
substantial extension of the implication of the eight-dimensional hidden
conformal symmetry, which goes far beyond the Mellin poles associated with the
leading logarithmic singularities. This leaves only a small finite set of poles
which can be determined on a case-by-case basis from the contributions of
protected operators in the OPE.
