In what sense are Pauli matrices "intertwiners"? I'm trying to understand the following quote from the Wikipedia article on the Pauli matrices.

More formally, this defines a map from $\mathbb {R} ^{3}$ to the vector space of traceless Hermitian $2\times 2$ matrices. This map encodes structures of $\mathbb {R} ^{3}$ as a normed vector space and as a Lie algebra (with the cross-product as its Lie bracket) via functions of matrices, making the map an isomorphism of Lie algebras. This makes the Pauli matrices intertwiners from the point of view of representation theory.

I understand everything except for the last sentence. I know what an intertwining map is, but what is an "intertwiner"? More importantly, the definition of an intertwining map $\phi:V\to W$ requires that we have linear representations (of the Lie algebra $\frak{so}(3)$, in this case) on $V$ and $W$. I understand that $\mathbb R^3$ with the cross product is isomorphic to $\mathfrak{so}(3)$, but this is not a typical linear representation, where the representatives of Lie algebra elements are linear maps and $\rho([X,Y])=\rho(X)\rho(Y)-\rho(Y)\rho(X)$. So what does the last sentence mean exactly?
 A: I'm pretty sure that your objection is exactly right. That is, the Pauli matrices are literally a/the standard basis for the (complexified?!) Lie algebra $\mathfrak sl(2)$. Yes, $\mathbb R^3$ with cross product is another 3-dimensional Lie algebra, and (maybe complexified?) is isomorphic to $\mathfrak sl(2)$.
I'd call the map a "Lie algebra isomorphism".
As you say, there are not really any representations in sight, and an "intertwiner" would usually be a map from one repn (of a fixed thing) to another repn (of that same thing), that preserves/respects the action (of the fixed thing).
So, yes, I agree with your appraisal that the terminology "intertwiner" in that situation is not standard, at least in the mathematics that I know, or the physics-y stuff that I know, either. But, still, conceivably, there is another milieu in which this abuse of terminology is standard, too.
A: A Lie group operates on its Lie algebra via the adjoint representation which is the conjugation for matrix Lie algebras. This means in case of unitary matrices
\begin{align*}
\operatorname{Ad}\, : \,\operatorname{SU}(2,\mathbb{C})&\longrightarrow \operatorname{SO}(\mathfrak{su}_\mathbb{R}(2,\mathbb{C)},\mathbb{R})\cong \operatorname{SO}(3,\mathbb{R})\\
\operatorname{Ad}\, : \,u&\longmapsto (X\longmapsto u^{-1}Xu)
\end{align*}
The Pauli matrices, however,
\begin{equation} \sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\, , \, \sigma_2 = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\, , \, \sigma_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \end{equation}
are hermitian, not skew-hermitian as required!
We either have to make them skew-hermitian by substituting them with $\{i\cdot \sigma_1,i\cdot \sigma_2,i\cdot \sigma_3\}$ as a basis of
$$
\mathfrak{su}(2,\mathbb{C})=\{X \in \mathbb{M}(2,\mathbb{C})\,|\,\operatorname{trace}X=0\wedge X+\overline{X}^T=0\}
$$
or drop the requirement that the representation space is a Lie algebra:
$$
\sigma_k + \overline{\sigma_k}^T=2\sigma_k\quad , \quad(i\sigma_k)+\overline{(i\sigma_k)}^T=0
$$
