  In this note we refine the notion of conditionally negative definite kernels
to the notion of conditionally strictly negative definite kernels and study its
properties. We show that the class of these kernels carries some surprising
rigidity, in particular, the word metric function on Coxeter groups is
conditionally strictly negative definite if and only if the group is a free
product of a number of copies of $\mathbb{Z}_2$'s and that the class of
conditionally strictly negative definite kernels on a finite set is a
one-parameter perturbation of the class of strictly positive definite kernels
on this set. We also discuss several examples.
