  We illustrate a rigorous approach to express the totally symmetric isotropic
tensors of arbitrary rank in the $n$-dimensional Euclidean space as a linear
combination of products of Kronecker deltas. By making full use of the
symmetries, one can greatly reduce the efforts to compute cumbersome angular
integrals into straightforward combinatoric counts. This method is generalized
into the cases in which such symmetries are present in subspaces. We further
demonstrate the mechanism of the tensor-integral reduction that is widely used
in various physics problems such as perturbative calculations of the
gauge-field theory in which divergent integrals are regularized in
$d=4-2\epsilon$ space-time dimensions. The main derivation is given in the
$n$-dimensional Euclidean space. The generalization of the result to the
Minkowski space is also discussed in order to provide graduate students and
researchers with techniques of tensor-integral reduction for particle physics
problems.
