  We propose a unified approach to the study of isometries on algebras of
vector-valued Lipschitz maps and those of continuously differentiable maps by
means of the notion of admissible quadruples. We describe isometries on
function spaces of some admissible quadruples that take values in unital
commutative $C^*$-algebras. As a consequence we confirm the statement of
\cite[Example 8]{jp} on Lipschitz algebras and show that isometries on such
algebras indeed take the canonical form.
