  Integrable discrete scalar equations defined on a~two or a three dimensional
lattice can be rewritten as difference systems in bond variables or in face
variables respectively. Both the difference systems in bond variables and the
difference systems in face variables can be regarded as vector versions of the
original equations. As a result, we link some of the discrete equations by
difference substitutions and reveal the non-potential versions of some
consistent-around-the-cube equations. We obtain higher-point configurations,
including pairs of compatible six~points equations on the ${\mathbb Z}^2$
lattice together with associated seven points equations. Also we obtain a
variety of compatible ten point equations together with associated ten and
twelve point equations on the ${\mathbb Z}^3$ lattice. Finally, we present
integrable multiquadratic quad relations.
