Exercise 2, § 3

If $f$ is a left-invariant pseudometric, the real-valued function $g(x) = f(e, x)$ on $G$ ($e$ being the identity element of $G$) satisfies the following conditions:

(i) $g(x) > 0$ for all $x \in G$, and $g(e) = 0$; (ii) $g(x^{-1}) = g(x)$; (iii) $g(xy) \leq g(x) + g(y)$.

Conversely, if $g$ is any real-valued function on $G$ satisfying these conditions, then $f(x, y) = g(x^{-1}y)$ is a left-invariant pseudometric on $G$.