After much shifting through notational hurdles, I may have gotten the point. However, I'd like to confirm this unequivocally by working through an example.
If $\beta \in V^*$ is $\beta=\begin{bmatrix}1 &2 &3 \end{bmatrix}$ and $\gamma\in V^*$ is $\gamma=\begin{bmatrix}2 &4 &6 \end{bmatrix}$. The $(2,0)$-tensor $\beta\otimes \gamma$ is the outer product:
$$\beta\otimes_o \gamma=\begin{bmatrix}2\,e^1\otimes e^1&4\,e^1\otimes e^2&6\,e^1\otimes e^3\\4\,e^2\otimes e^1&8\,e^2\otimes e^2&12\,e^2\otimes e^3\\6\,e^3\otimes e^1&12\,e^3\otimes e^2&18\,e^3\otimes e^3\end{bmatrix}$$
Now if apply this tensor product on the vectors
$$v=\begin{bmatrix}1\\-1\\5\end{bmatrix}, \; w = \begin{bmatrix}2\\0\\3\end{bmatrix}$$
$$\begin{align} (\beta \otimes \gamma)[v,w]&=\\[2ex] & 2 \times 1 \times 2 \quad+\quad 4 \times 1 \times 0 \quad +\quad 6 \times 1 \times 3 \\ +\;&4 \times -1 \times 2 \quad + \quad 8 \times -1 \times 0 \quad + \quad 12 \times -1 \times 3 \\ +\;&6 \times 5 \times 2 \quad + \quad 12 \times 5 \times 0 \quad + \quad 18 \times 5 \times 3 \\[2ex] &= 308\end{align}$$
Is this correct?
v = c(1,-1,5); w = c(2,0,3); beta = 1:3; gamma = c(2,4,6); beta %*% v * gamma %*% w 308.
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