  We design a Fortin operator for the lowest-order Taylor-Hood element in any
dimension, which was previously constructed only in 2D. In the construction we
use tangential edge bubble functions for the divergence correcting operator.
This naturally leads to an alternative inf-sup stable reduced finite element
pair. Furthermore, we provide a counterexample to the inf-sup stability and
hence to existence of a Fortin operator for the $P_2$-$P_0$ and the augmented
Taylor-Hood element in 3D.
