Triple parabolic cosets have unique Bruhat minima

A folklore result says that any parabolic double coset $W_{I}w{W}_J$ in a Coxeter group $W$ is an interval of the Bruhat order, provided that $W_I$ and $W_J$ are finite. We include a self-contained proof, without assuming that $W$ is finite. Furthermore, we prove that each “triple coset” $W_{I}u{W}_{J}v{W}_K$ has a Bruhat minimum and – if $W_I,W_J,W_K$ are finite – a Bruhat maximum, but (in general) is not a Bruhat interval. This proves and extends a conjecture on MathOverflow #472072.