Are Pauli matrices invariant tensors in the representation of $\frac12 \otimes \frac12 \otimes 1$?

Question

If we raise the index of the Pauli matrices with Levi-Civita symbol $\epsilon$ we obtain the 2-index spinors $(\sigma_i)^{AB} = (\sigma_i)^A{}_C \ \epsilon^{CB}$. The textbook (Ref. 1) argued that they are invariant tensors in the representation $\frac12 \otimes \frac12 \otimes 1$. However, I raised the index by $\epsilon$, which gives that: $$ (\sigma_1)^{AB}=\left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \\ \end{array} \right), \quad (\sigma_2)^{AB}=\left( \begin{array}{cc} i & 0 \\ 0 & i \\ \end{array} \right),\quad (\sigma_3)^{AB}=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right). $$ How can I verify they are invariant under the representation $\frac12 \otimes \frac12 \otimes 1$?

Reference:

1. Carlo Rovelli and Francesca Vidotto, Covariant Loop Quantum Gravity, Exercise 1.9 pp. 27. ISBN:9781107069626.

Answer 1

Yes, the Pauli matrices are invariant in the sense that $$\sum_{j=1}^3 U \sigma_j U^{-1}(R^{-1})^j{}_k~=~ \sigma_k ,\tag{A} $$ where $U\in SU(2)$ is a $2\times 2$-matrix in the spin-$1/2$ representation and $R\in SO(3)$ is the corresponding $3\times 3$-matrix in the spin-$1$ representation, cf. e.g. my Phys.SE answer here.