I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/).

There comes a point in the paper (Lemma 2.8) when he uses the fact that the ring of all constant coefficient $SO(n)$-invariant differential operators on $M_{n,m}$ (viewed as $\mathbb R^{nm}$) without constant term is generated by the operators $$\Delta_{il}=\sum_{k=1}^{n} \frac{\partial}{\partial x_{ki}}\frac{\partial}{\partial x_{kl}} \;\;\;\;\;\;\;1 \leq i, l \leq m$$ where $m < n$.

I don't see how the ring is generated by operators of this form. How can we show this?