  Let $A$ be a $\nu$-vector of self-adjoint, pairwise commuting operators and
$B$ a bounded operator of class $C^{n_0}(A)$. We prove a Taylor-like expansion
of the commutator $[B,f(A)]$ for a large class of functions
$f\colon\mathbm{R}^\nu \to \mathbm{R}$, generalising the one-dimensional result
where $A$ is just a self-adjoint operator. This is done using almost analytic
extensions and the higher-dimensional Helffer-Sj\"ostrand formula.
