Let $V=\mathbb{C}^d$ be a $d$-dimensional vector space over $\mathbb{C}$. The problem is to find all possible representations $\rho:\mathfrak{l}\to \mathrm{End}(V^3)$ of the Lie algebra $\mathfrak{l}=\mathfrak{su}(2)$ in the product space $V^3=V\otimes V\otimes V$ with the following property:
(*)There exists two elements $l_1,l_2\in \mathfrak{l}$ such that $\rho(l_1)=A_{12}+C\equiv A\otimes \mathbb{1}+C,\rho(l_2)=A_{23}+C\equiv \mathbb{1}\otimes A +C$, where $A\in \mathrm{End}(V^2)$ (i.e. acts in $V\otimes V$), $\mathbb{1}$ is the identity operator in $\mathrm{End}(V)$, and $A_{12},A_{23}$ denotes two different ways of embedding of $A$ in $\mathrm{End}(V^3)$. $C\in \mathrm{End}(V^3)$ is a suitable "central element" that commutes with $\rho(l)$ for $\forall l\in \mathfrak{l}$.
Examples:
(1). For $d$=2, consider the representation $\rho(s^x)=\sigma^z_1 \sigma^x_2/2,\rho(s^z)=\sigma^z_2 \sigma^x_3/2,\rho(s^y)=\sigma^z_1 \sigma^y_2\sigma^x_3/2$, where $\sigma^x,\sigma^y,\sigma^z$ are Pauli matrices. This is the 8-dimensional direct sum representation $\frac{1}{2}\oplus\frac{1}{2}\oplus\frac{1}{2}\oplus\frac{1}{2}$. We have $l_1=s^x,l_2=s^z,A=\sigma^z\otimes\sigma^x$ (or $A_{12}=\sigma^z_1\sigma^x_2,A_{23}=\sigma^z_2\sigma^x_3$), and $C=0$. So this representation satisfies the above property(*).
(2) For arbitrary $d$, consider the representation \begin{eqnarray} \rho(s^x)=\frac{2P_{12}-P_{23}-P_{13}}{3\sqrt{6}}+\frac{M}{6i}\\ \rho(s^y)=\frac{2P_{23}-P_{13}-P_{12}}{3\sqrt{6}}+\frac{M}{6i}\\ \rho(s^z)=\frac{2P_{13}-P_{12}-P_{23}}{3\sqrt{6}}+\frac{M}{6i}, \end{eqnarray} where $P\in\mathrm{End}(V^2)$ is the permutation operator defined by $P(e_1\otimes e_2)=e_2\otimes e_1$ for $\forall e_1,e_2\in V$, and $P_{12},P_{23},P_{13}$ are the different embedding of $P$ in $\mathrm{End}(V^3)$, e.g. $P_{13}(e_1\otimes e_2\otimes e_3)=(e_3\otimes e_2\otimes e_1)$, etc., and $M=[P_{12},P_{23}]=P_{12}P_{23}-P_{23}P_{12}=[P_{23},P_{13}]=[P_{13},P_{12}]$. Put it more formally, $\rho$ maps $\mathfrak{su}(2)$ to the group algebra $\mathbb{C}(S_3)$ of the symmetry group $S_3$. Therefore, let $l_1=\frac{2s^x-s^y-s^z}{\sqrt{6}},l_2=\frac{2s^y-s^z-s^x}{\sqrt{6}}$, we have $$\rho(l_1)=\frac{P_{12}}{2}-\frac{c}{6},~~~\rho(l_2)=\frac{P_{23}}{2}-\frac{c}{6},$$ where $c=P_{12}+P_{23}+P_{13}$ is a central element of $\mathbb{C}(S_3)$ and therefore commutes with $\rho(l)$ for $\forall l\in \mathfrak{su}(2)$. Therefore, this representation satisfies the property(*) with $A=P/2$ and $C=-c/6$. When $d=2$, this representation is the 8-dimensional direct sum representation $0^4\oplus \frac{1}{2}\oplus\frac{1}{2}$.
The above two are all the examples I know up to now. Is there a systematic way of finding all such representations? Interestingly, notice the angle between $l_1$ and $l_2$: in example(1), $\angle(l_1,l_2)=\pi/2$, in example(2), $\angle(l_1,l_2)=2\pi/3$. What are all possible allowed angles between $l_1$ and $l_2$ (infinitely many or finitely many)? Or are $\pi/2,2\pi/3$ the only allowed values of $\angle(l_1,l_2)$?
Here is an alternative formulation of the same problem: find all operators $A\in \mathrm{End}(V^2)$ such that the set of operators $\{A_{12}+C,A_{23}+C\}$ in $\mathrm{End}(V^3)$ generates the Lie algebra $\mathfrak{su}(2)$, for a suitable $C\in \mathrm{End}(V^3)$ that commutes with $A_{12},A_{23}$. In example (1) $A=\sigma^z\otimes\sigma^x,C=0$ and in example (2) $A=P/2,C=-(P_{12}+P_{23}+P_{13})/6$.
Any hints, references, or further examples will be welcomed. Thanks!