In several situations in differential geometry, one can be interested in determining all inner products on a vector space that are invariant under a given group action. For example, bi-invariant Riemannian metrics on a Lie group G are characterized by Ad(G)-invariant inner products on the Lie algebra g. Analogously, G-invariant Riemannian metrics on a homogeneous space M = G/H are characterized by Ad(H)invariant inner products on the tangent space TH M. In addition, given a G-equivariant diffeomorphism between a manifold M and a Euclidean space V , G-invariant log-Euclidean metrics can be defined on M by pullback of G-invariant inner products on V. There exists a general procedure based on representation theory to find all invariant inner products on a completely reducible Hermitian space. It consists in changing the viewpoint from invariant inner products to equivariant automorphisms. The goal of this work is to diffuse this method to communities of applied mathematics which use differential geometry. Therefore, in this work, we recall this general method that we did not find elsewhere, along with an elementary presentation of the basics of representation theory.