The questions I pose here concerns the paper "The Spin Model of Euclidean 3-Space" by W. F. Eberlein (The American Mathematical Monthly, Vol. 69, No. 7 (Aug. - Sep., 1962), pp. 587-598) (download or free online reading)
Since there is a typo in the paper just in the place I want to analyze, it's important to note that the second equation on page 597 shoud read as follows: \begin{align} i\hbar I\frac{\partial\psi}{\partial t}=& \left[\frac{1}{2m}\left(-i\hbar\nabla-\frac{e\mathfrak A}{c}\right)^2+e\Phi I\right]\psi\\=& \left[\frac{-\hbar^2}{2m}\nabla^2+\frac{ie\hbar}{2mc}(\nabla\mathfrak A+\mathfrak A\nabla)+\left(\frac{e^2\mathfrak A^2}{2mc^2}+e\Phi I\right)\right]\psi \end{align} , that is, $\nabla\cdot\mathfrak A+\mathfrak A\cdot\nabla$ in the paper should read as $\nabla\mathfrak A+\mathfrak A\nabla$ in this equation (see Eberlein, W. F. “A Crucial Correction.” The American Mathematical Monthly 69, no. 10 (1962): 960–960. download or free online reading).
Background
The "spin model of the 3-dimensional Euclidean space" is the 3-dimensional Euclidean space $\frak E_3$ spanned by the Pauli-matrices, i.e, it is the real vector space of the Hermitian and traceless complex $2\times 2$ matrices. It is an Euclidean space with respect to the scalar product $\cdot$ defined by the following equation: $$\frac{1}{2}(AB+BA)=(A\cdot B)I\quad (A,B\in \mathfrak E_3)\tag 1$$ where $I$ is the $2\times 2$ identity matrix. This definition is based on the fact that for all $A, B \in\mathfrak E_3$, $AB+BA = \delta_{ij}kI$, where $k$ is real. The positive definiteness is ensured by the $\sigma_i\sigma_j+\sigma_2\sigma_i=\delta_{ij}I$ anticommutation relations of the Pauli matrices $\sigma_i$ and $\sigma_j$. These commutation relations also show that Pauli matrices form an orthonormal basis of $\mathfrak E_3$. An important property of $\mathfrak E_3$ as a subset of $M_{2\times 2}(\mathbb C)$ that $$M_{2\times 2}(\mathbb C)=\mathrm{span}{(\{I\})}\oplus\mathfrak E_3\oplus i\mathfrak E_3\oplus i\,\mathrm{span}(\{I\}).\tag 2$$ In his paper, Eberlein considers $\mathbb C^n$ endowed with the Hermitian product $(x,y)=\sum_1^n x_j\overline y_j$ and denotes this Hermitian space by $H_n$, while he regards $M_{n\times n}$ as the algebra of the linear maps of $H_n$ to itself, and denotes it by $B(H_n)$.
Multiplications by differential operators
I restrict my question to the case a) of the paper, i.e. when $\psi:\mathfrak E_3\to \mathbb C$.
Taking an orthonormal base $\{\xi_1,\xi_2,\xi_3\}$ of $\mathfrak E_3$, Eberlein defines the symbol $\nabla$ only formally, as $\nabla = \xi_1\partial_1+\xi_2\partial_2+\xi_3\partial_3$ and $\nabla\psi$ as $$\nabla\psi=\xi_1(\partial_1\psi)+\xi_2(\partial_2\psi)+\psi_3(\partial_3\psi)\tag i.$$ I guess he means the function $$\nabla\psi:\mathfrak E_3\to B(H_2):p\mapsto\partial_1\psi(p)\xi_1+\partial_2\psi(p)\xi_2+\partial_3\psi(p)\xi_3,\tag{ii}$$ that is, the value of $\nabla\psi$ at a point $p\in\mathfrak E_3$ is a $2\times 2$ complex matrix with trace = $0$, or in other words, it is an element of $\mathfrak E_3\oplus i\mathfrak E_3$.
Taking $\mathfrak A=\mathfrak A_1\xi_1+\mathfrak A_2\xi_2+\mathfrak A_3\xi_3\in\mathfrak E_3$ ($\mathfrak A_i\in\mathbb R$), the product $\nabla\mathfrak A$ is the formal matrix product of $\nabla$ and $\mathfrak A$, that is,
$$\nabla\mathfrak A=\sum_{i,j=1}^3\xi_i\partial_i\mathfrak A_j\xi_j\tag{iii}$$ where the $\xi_i$-s and $\xi_j$-s are regarded as $2 \times 2$ complex matrices, and the product $\mathfrak A\nabla$ is the formal matrix product of $\mathfrak A$ and $\nabla$ , that is, $$\mathfrak A\nabla=\sum_{i,j=1}^3\mathfrak A_j\xi_j\xi_i\partial_i\tag{iv},$$
while the product $\nabla\cdot\mathfrak A$ is the formal dot product of $\nabla$ and $\mathfrak A$, that is, $$\nabla\cdot\mathfrak A=\partial_1\mathfrak A_1+\partial_2\mathfrak A_2+\partial_3\mathfrak U_3\tag{v}$$ and $\mathfrak A\cdot \nabla$ is the formal dot product of $\nabla$ and $\mathfrak A$, that is, $$\mathfrak A\cdot\nabla=\mathfrak A_1\partial_1+\mathfrak A_2\partial_2+\mathfrak A_3\partial_3\tag{vi}$$
From the derivation of eq. (8) in the paper, one sees that according to the author, in an orthonormal basis $\{\xi_1,\xi_2,\xi_3\}$ of $\mathfrak E_3$, $$\begin{align}\nabla \mathfrak A+\mathfrak A\nabla = & (\nabla \cdot \mathfrak A+\mathfrak A\cdot \nabla)I \\ & \color{red}{+i\left[\xi_1\{\partial_2\mathfrak A_3- \partial_3\mathfrak A_2\}+\xi_2\{\partial_3\mathfrak A_1- \partial_1\mathfrak A_3\}+\xi_3\{\partial_1\mathfrak A_2- \partial_2\mathfrak A_1\}\right]}. \end{align}\tag 3$$ Since every definition is formal, I would expect any identities of $\mathfrak E_3$ remain valid also for the formal $\nabla$ symbol too. (however, the comment by Kurt G. disproves my expectations ab ovo), so I would expect, according to (1), that $$\nabla \mathfrak A+\mathfrak A\nabla = (\nabla \cdot \mathfrak A+\mathfrak A\cdot \nabla)I \tag 4.$$
As the calculations in the paper are very formal, I cannot say whether this is a real contradiction or an apparent one. Could someone please explain in clear mathematical terms the cause of the difference between (3) and (4)?
