  A pseudo-length function defined on an arbitrary group $G = (G,\cdot,e,
(\,)^{-1})$ is a map $\ell: G \to [0,+\infty)$ obeying $\ell(e)=0$, the
symmetry property $\ell(x^{-1}) = \ell(x)$, and the triangle inequality
$\ell(xy) \leqslant \ell(x) + \ell(y)$ for all $x,y \in G$. We consider
pseudo-length functions which saturate the triangle inequality whenever $x=y$,
or equivalently those that are homogeneous in the sense that $\ell(x^n) =
n\,\ell(x)$ for all $n\in\mathbb{N}$. We show that this implies that
$\ell([x,y])=0$ for all $x,y \in G$. This leads to a classification of such
pseudo-length functions as pullbacks from embeddings into a Banach space. We
also obtain a quantitative version of our main result which allows for defects
in the triangle inequality or the homogeneity property.
