We introduce a general definition of homogeneous Besov spaces on a stratified
Lie group $G$, based on a Littlewood-Paley-type decomposition of Schwartz
functions with all moments vanishing. We show that under mild and intuitive
conditions the spaces thus defined are independent of the decomposition
employed. A corollary of this is that previously constructed versions of
homogeneous Besov spaces on $G$, relying on the spectral calculus of a
sub-Laplacian of the group, are consistent, i.e., independent of the choice of
sub-Laplacian.

We further prove characterizations of homogeneous Besov spaces using
continuous wavelet transforms, with a large variety of analysing wavelets to
choose from.