Geometric explanation of Fueter-Sce-Qian theorem and similar situations

Question

In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}_m$, $\ $ $\Delta^{\frac{m-1}{2}}f$ is a monogenic function, namely $$ \Delta^{\frac{m-1}{2}}f\in\ker{\overline{\partial}}, $$ where $\overline{\partial}:=\partial_{x_0}+\sum_{j=1}^me_j\partial_{x_j}$ is the Dirac operator. For simplicity, think of slice regular functions as a polynomials in the paravector variable $P:x=x_0+ \sum_{j=1}^me_jx_j\mapsto P(x)=\sum_{j=1}^nx^ja_j$.
The exponent $\frac{m-1}{2}$ is somehow critical, indeed, given a polynomial $P=\sum_{j=1}^nx^ja_j$ we have for $k<\frac{m-1}{2}$ $$ \overline{\partial}\Delta^kP=0\iff n\leq2k, $$ while for $k\geq\frac{m-1}{2}$, $$ \overline{\partial}\Delta^kP=0\quad \forall n\in\mathbb{N}. $$ Thus, if we study the function $\mathcal{F}_k:=\overline\partial\Delta^k$ restricted to polynomials in the paravector variable $\mathbb{R}^{m+1}$ we have a chain of inclusions of kernels that blows up in $\frac{m-1}{2}$: $$ \ker{\mathcal{F}_0}=\mathbb{R}^{m+1}_0[x]\subset\ker{\mathcal{F}_1}=\mathbb{R}^{m+1}_2[x]\subset\dots\subset\ker{\mathcal{F}_\frac{m-3}{2}}=\mathbb{R}^{m+1}_{m-3}[x]\subset\ker{\mathcal{F}_\frac{m-1}{2}}=\mathbb{R}^{m+1}[x], $$ where $\mathbb{R}^{m+1}_j[x]$ means the set of polynomials of degree less or equal than $j$.
Do you know if there are similar casis or a more general situation in which this phenomenon happens, or are there geometric reasons for this to happen?