intertwiner acts on the Hilbert space $\CS_j\otimes\CS'_j$, the tensor product of two carrier spaces of the same $j$. In general we wish to consider the second carrier space as distinct from the first, which is the purpose of the prime on the second factor. To within a normalization and phase, there is a unique vector in this space that is invariant under rotations; we call it $\ket{K}$, and express it in terms of the Clebsch-Gordan coefficients by $$\ket{K}= \sqrt{2j+1} \sum_{mm'} \ket{jm}\otimes\ket{jm'}\,
     C^{00}_{jjmm'}.
    \label{ketKdef}$$ This vector can also be expressed in terms of the “$2j$-symbol,” which we define in terms of the usual $3j$-symbol by $$\fl\left(\begin{array}{cc}
        j & j \\
        m & m'
    \end{array}\right)=
    \left(\begin{array}{ccc}
    j & j & 0 \\
    m & m' & 0
    \end{array}\right) = 
    C^{00}_{jjmm'} =
    \frac{(-1)^{j-m}}{\sqrt{2j+1}}
    \,\delta_{m,-m'}.
    \label{2jdef}$$ The terminology “$2j$-symbol” is not entirely standard, but it has been used by Stedman (1990). The invariant vector $\ket{K}$ can also be written, $$\fl\ket{K}=\sum_{mm'} \ket{jm}\otimes\ket{jm'}\,
    (-1)^{j-m} \,\delta_{m,-m'} = \sum_m
    \ket{jm}\otimes\ket{j,-m}\,(-1)^{j-m}.
    \label{K2jdef}$$

By Hermitian conjugation we convert $\ket{K}$ into the bra $\bra{K}$, which is otherwise an intertwiner $K:\CS_j\otimes\CS'_j\to\Complexes$. Just as the components of the intertwiner $W$ are the $3j$-symbol, the components of the intertwiner $K$ are the $2j$-symbol, multiplied 