Def 1. A distance $d$ defined on a set $\mathcal X \times \mathcal X$ is said to be Hilbertian if there exists a Hilbert space $\mathcal H$ and a mapping $\phi: \mathcal X \to \mathcal H$ such that for any pair $x, x' \in \mathcal X$ we have $d(x,x') = ||\phi(x) - \phi(x')||_{\mathcal H}$.
Def 2. A symmetric function $k$ defined on a set $\mathcal X \times \mathcal X$ is said positive definite if for any $n \geq 0$, the familiy $x_1,...,x_n \in \mathcal X$ and vector $r \in \mathbb R^n$ the following inequality holds: $$ \sum^n_{i,j = 1} r_i r_j k(x_i,x_j) \geq 0 $$ The negative definite is just the opposite inequality. Finally, we call $k$ conditionally positive if this inequality if valid only for $r$ such that $\sum_i r_i = 0$.
Now, on the book Computational Optimal Transport, the authors claim that a distance if $d$ is Hilbertian, then $d^2$ is negative definite. This claim is said to be trivial, but I don't see how this follows. How can one show that indeed $d^2$ is negative definite?