For \(d=3\) there is a proof involving analytic tools in [1, Lemma 17.13].

For harmonic polynomials of a certain degree the inner product can be defined combinatorially, by identifying the left polynomial with a differential operator [2].

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  1. B. Hall, Quantum Theory for Mathematicians. New York, NY: Springer New York, 2013. doi:10.1007/978-1-4614-7116-5
  2. J. Gallier. Class Lecture, Topic: Notes on Spherical Harmonics and Linear Representations of Lie Groups.